The true identity of Young Griff is often discussed as to who exactly he is. There seems to be several prevalent theories that I will try and mention here but the main goal is just to list all of the "potential" evidence/foreshadowing available and leave it open to discussion.

Form Your Own Opinion: Who Is Young Griff?

As I mentioned numerous theories exist with the focus primarily on:

• Aegon Blackfyre (a descendant of the female line)
• Aegon VI Targaryen (son of Rhaegar/Elia)
• Pretender (just some guy with Valyrian features)
• Son of Illyrio/Serra

There are other theories too, but far less evidence (descendant of Aerion Brightflame's son Maegor, Shiera Seastar, etc.)

Note: Numerous of these quotes could support more than one character, it was too much of a headache to list them under everyone, but I readily recognize that

Note II: I am going to try hard to approach this post with as little bias as possible, but I will admit that I readily think that Young Griff is a Blackfyre.

Background

As far back as as A Storm of Swords release GRRM was already dropping hints that this character would survive in some form:

I was wondering if you could answer (or take the "fifth") one teeny little question I've been dying to ask for the past year***: Are Aegon and Rhaenys, Elia's children, well and truly dead****?*

All I have to say is that there is absolutely no doubt that little Princess Rhaenys was dragged from beneath her father's bed and slain. -SSM, Elia's Children: 6 Aug 2000

Aegon VI Targaryen

The story the reader gets from Young Griff is that someone else died in his place:

The lad flushed. "That was not me. I told you. That was some tanner's son from Pisswater Bend whose mother died birthing him. His father sold him to Lord Varys for a jug of Arbor gold. He had other sons but had never tasted Arbor gold. Varys gave the Pisswater boy to my lady mother and carried me away."

"Aye." Tyrion moved his elephants. "And when the pisswater prince was safely dead, the eunuch smuggled you across the narrow sea to his fat friend the cheesemonger, who hid you on a poleboat and found an exile lord willing to call himself your father. It does make for a splendid story, and the singers will make much of your escape once you take the Iron Throne … assuming that our fair Daenerys takes you for her consort." -ADWD, Tyrion VI

which is possible:

Kevan Lannister had been here, in this very hall when Tywin had laid the bodies of Prince Rhaegar's children at the foot of the Iron Throne*, wrapped up in crimson cloaks. The girl had been recognizably the Princess Rhaenys, but the boy ... a faceless horror of bone and brain and gore, a few hanks of fair hair. None of us looked long. Tywin said that it was Prince Aegon, and we took him at his word. -ADWD, Epilogue

Dany sees Rhaegar/Elia/Aegon (confirmed by GRRM) in the House of the Undying:

Viserys, was her first thought the next time she paused, but a second glance told her otherwise. The man had her brother's hair, but he was taller, and his eyes were a dark indigo rather than lilac. "Aegon," he said to a woman nursing a newborn babe in a great wooden bed. "What better name for a king?"

"Will you make a song for him?" the woman asked.

"He has a song," the man replied. "He is the prince that was promised, and his is the song of ice and fire." He looked up when he said it and his eyes met Dany's, and it seemed as if he saw her standing there beyond the door. "There must be one more," he said, though whether he was speaking to her or the woman in the bed she could not say. "The dragon has three heads." He went to the window seat, picked up a harp, and ran his fingers lightly over its silvery strings. Sweet sadness filled the room as man and wife and babe faded like the morning mist, only the music lingering behind to speed her on her way. -ACOK, Daenerys IV

and:

She nodded. "There was a woman in a bed with a babe at her breast. My brother said the babe was the prince that was promised and told her to name him Aegon."

"Prince Aegon was Rhaegar's heir by Elia of Dorne," Ser Jorah said. "But if he was this prince that was promised, the promise was broken along with his skull when the Lannisters dashed his head against a wall." -ACOK, Daenerys V

Some potential foreshadowing:

The dragon has three heads. There are two men in the world who I can trust, if I can find them. I will not be alone then. We will be three against the world, like Aegon and his sisters. -ASOS, Daenerys VI

Daenerys' Thoughts on Aegon as a spouse:

Five Aegons had ruled the Seven Kingdoms of Westeros. There would have been a sixth, but the Usurper's dogs had murdered her brother's son when he was still a babe at the breast. If he had lived, I might have married him. Aegon would have been closer to my age than Viserys. Dany had only been conceived when Aegon and his sister were murdered. -ADWD, Daenerys I

One major argument is that Varys has no reason to lie to a dead man (that said he doesn't ever actually say Aegon is Aegon VI):

Doubt, division, and mistrust will eat the very ground beneath your boy king, whilst Aegon raises his banner above Storm's End and the lords of the realm gather round him."

"Aegon?" For a moment he did not understand. Then he remembered. A babe swaddled in a crimson cloak, the cloth stained with his blood and brains. "Dead. He's dead."

"No." The eunuch's voice seemed deeper. "He is here. Aegon has been shaped for rule since before he could walk. He has been trained in arms, as befits a knight to be, but that was not the end of his education. He reads and writes, he speaks several tongues, he has studied history and law and poetry. A septa has instructed him in the mysteries of the Faith since he was old enough to understand them. He has lived with fisherfolk, worked with his hands, swum in rivers and mended nets and learned to wash his own clothes at need. He can fish and cook and bind up a wound, he knows what it is like to be hungry, to be hunted, to be afraid. Tommen has been taught that kingship is his right. Aegon knows that kingship is his duty, that a king must put his people first, and live and rule for them."

Aegon Blackfyre

We are told multiple times that the Blackfyre line is only dead in the male line:

When Maelys the Monstrous died upon the Stepstones, it was the end of the male line of House Blackfyre." -ADWD, Tyrion II

and:

Daemon Blackfyre's surviving sons fled to Tyrosh, their mother's home, and with them went Bittersteel. The realm would continue to be troubled by the claims of the Blackfyre Pretenders for four more generations, until the last of the descendants of Daemon Blackfyre through the male line was sent to the grave. -TWOIAF, The Targaryen Kings: Daeron II

When asked why the Golden Company would support a Targaryen claimant:

"How did you convince the Golden Company to take up th-e cause of our sweet queen when they have spent so much of their history fighting against the Targaryens?"

Illyrio brushed away the objection as if it were a fly. "Black or red, a dragon is still a dragon. -ADWD, Tyrion II

The story of the black iron dragon being thrown into the sea and returning covered with rust could be a metaphor (black dragon returning to westeros disguised as a red dragon).

"Is the dragon sign still there?" asked Podrick.

"No," said Septon Meribald. "When the smith's son was an old man, a bastard son of the fourth Aegon rose up in rebellion against his trueborn brother and took for his sigil a black dragon. These lands belonged to Lord Darry then, and his lordship was fiercely loyal to the king. The sight of the black iron dragon made him wroth, so he cut down the post, hacked the sign into pieces, and cast them into the river. One of the dragon's heads washed up on the Quiet Isle many years later, though by that time it was red with rust. The innkeep never hung another sign, so men forgot the dragon and took to calling the place the River Inn. In those days, the Trident flowed beneath its back door, and half its rooms were built out over the water. Guests could throw a line out their window and catch trout, it's said. There was a ferry landing here as well, so travelers could cross to Lord Harroway's Town and Whitewalls." -AFFC, Brienne VII

If Illyrio does indeed have Blackfyre (as potentially seen in early versions of Tyrion's ADWD chapters with a gift that tyrion hears "sword" that was removed from the published version) it would potentially mean A LOT as that sword carries a ton of weight:

[Did Daena complaining about how she might have been Queen if it weren't for the Dance of the Dragons determining that a Targaryen queen would never rule in her own right lead to Daemon Blackfyre's rebellion?]

Certainly possible, but it was Aegon's very public gift of Blackfyre to his bastard son that first started widespread talk that perhaps he should be king. -SSM, Targaryen History: 21 April 2008

and:

Targaryen bastards have been given a number of different names over the years. The Blackfyres are one specific branch, descended from Daemon Blackfyre, a bastard son of Aegon IV the Unworthy by one of the three sisters that Baelor the Blessed imprisoned in the Maidenvault. Blackfyre was also the name of Aegon the Conquerer's greatsword, a fabled blade of Valyrian steel passed from king to king... until Aegon IV chose to bestow it on Daemon instead of his legitimate son, Daeron, whom he suspected was actually fathered by his brother, Prince Aemon the Dragonknight. Some felt that the sword symbolized the monarchy, so the gift was the seed from which the Blackfyre Rebellions grew. -SSM, SF, Targaryens, Valyria, Sansa, Martells: 26 June 2001

These quotes can easily be interpreted several ways (even not about Aegon)

but going back to Dany's HOTU vision (under the slayer of lies section):

A cloth dragon swayed on poles amidst a cheering crowd. -ACOK, Daenerys IV

and:

"A dead man in the prow of a ship, a blue rose, a banquet of blood . . . what does any of it mean, Khaleesi? A mummer's dragon, you said. What is a mummer's dragon, pray?"

"A cloth dragon on poles," Dany explained. "Mummers use them in their follies, to give the heroes something to fight." -ACOK, Daenerys V

and:

"No. Hear me, Daenerys Targaryen. The glass candles are burning. Soon comes the pale mare, and after her the others. Kraken and dark flame, lion and griffin, the sun's son and the mummer's dragon. Trust none of them. Remember the Undying. Beware the perfumed seneschal."

and:

"Dragons," Moqorro said in the Common Tongue of Westeros. He spoke it very well, with hardly a trace of accent. No doubt that was one reason the high priest Benerro had chosen him to bring the faith of R'hllor to Daenerys Targaryen. "Dragons old and young, true and false, bright and dark. -ADWD, Tyrion VIII

Pretender

Its quite possible that they just found a kid with valyrian looks and are using him:

The captain-general looked as if someone had slapped his face. "Has the sun curdled your brains, Flowers? We need the girl. We need the marriage. If Daenerys accepts our princeling and takes him for her consort, the Seven Kingdoms will do the same. Without her, the lords will only mock his claim and brand him a fraud and a pretender. - ADWD, The Lost Lord

and:

"Connington may have more than the Golden Company. It is said he has a Targaryen pretender."

"A feigned boy is what he has," said Randyll Tarly.

...

If she should reach these shores and join her strength to Lord Connington and this prince of his, feigned or no … we must destroy Connington and his pretender now, before Daenerys Stormborn can come west."

...

"Sellswords fight for coin," declared Grand Maester Pycelle. "With enough gold, we might persuade the Golden Company to hand over Lord Connington and the pretender." -ADWD, Epilogue

and:

Aurane did not resemble Prince Rhaegar as much as she had thought. He has the hair, but so do half the whores in Lys, if the tales are true. -AFFC, Cersei VIII

Illyrio/Serra's Son

Illyrio fell in love with one of his bedwarmers from a Lysene pillow house:

llyrio thrust his right hand up his left sleeve and drew out a silver locket. Inside was a painted likeness of a woman with big blue eyes and pale golden hair streaked by silver. "Serra. I found her in a Lysene pillow house and brought her home to warm my bed, but in the end I wed her. Me, whose first wife had been a cousin of the Prince of Pentos. The palace gates were closed to me thereafter, but I did not care. The price was small enough, for Serra." -ADWD, Tyrion II

She died from greyscale:

"Good fortune," Illyrio called after them. "Tell the boy I am sorry that I will not be with him for his wedding. I will rejoin you in Westeros. That I swear, by my sweet Serra's hands." -ADWD, Tyrion III

Wrt to Serra there are plenty of theories about her identity that I don't want to get into here as it would make the post way too long, but it should be noted that Shiera Seastar (the only* Great Bastard with no true effect on the plot has no fate yet either, although she is dead, we don't know of children, etc.)

SERENEI OF LYS (SWEET SERENEI): A Lysene beauty from an ancient but impoverished line, brought to court by Lord Jon Hightower, the new Hand. Serenei was the most beautiful of Aegon's mistresses, but she was also reputed to be a sorceress. She died giving birth to the last of the king's bastard children, a girl called Shiera Seastar who became the greatest beauty in the Seven Kingdoms, beloved of both her half brothers, Bittersteel and Bloodraven, whose rivalry would ripen to hatred. Children by Serenei: Shiera -TWOIAF, The Targaryen Kings: Aegon IV

Other quotes

These didn't necessarily fit anywhere but I like them:

His fellow drinkers were talking about dragons now. "You're bloody mad," said an oarsman off Storm Dancer. "The Beggar King's been dead for years. Some Dothraki horselord cut his head off."

"So they tell us," said the old fellow. "Might be they're lying, though. He died half a world away, if he died at all. Who's to say? If a king wanted me dead, might be I'd oblige him and pretend to be a corpse. None of us has ever seen his body."

"I never saw Joffrey's corpse, nor Robert's," growled the Eel's proprietor. "Maybe they're all alive as well. Maybe Baelor the Blessed's just been having him a little nap all these years." -ADWD, Davos II

and:

"Not Stannis. Nor Myrcella." The yellow smile widened. "Another. Stronger than Tommen, gentler than Stannis, with a better claim than the girl Myrcella. A savior come from across the sea to bind up the wounds of bleeding Westeros."

"A dragon." The cheesemonger saw the look on his face at that, and laughed. "A dragon with three heads." ADWD, Tyrion I

and:

When the lad emerged from the cabin with Lemore by his side, Griff looked him over carefully from head to heel. The prince wore sword and dagger, black boots polished to a high sheen, a black cloak lined with blood-red silk. With his hair washed and cut and freshly dyed a deep, dark blue, his eyes looked blue as well. At his throat he wore three huge square-cut rubies on a chain of black iron, a gift from Magister Illyrio. Red and black. Dragon colors. That was good. "You look a proper prince," he told the boy. "Your father would be proud if he could see you." -ADWD, The Lost Lord

Conclusion

Nothing really to conclude, I just attempted to list out most of the quotes having to do with Young Griff's identity. Some of the quotes can support more than one depending on how you look at them.

One thing I will note is that like R+L=J, we are never explicitly given the idea that Young Griff could be a Blackfyre in the series. As with R+L=J (Ashara, Dornish peasant,etc.) we are given numerous potential options by characters (Aegon VI, pretender, etc.) but a descendant of House Blackfyre has not been brought up once, it only exists because people pieced together clues.

It is also possible that more than one of the options above are true. Griff could be a descendant of House Blackfyre and Serra's child and so on.

Note: I was planning to link a ton of my posts on this subject but it made the post way too pro Blackfyre ish for my liking while this was supposed to be just a summary of different theories.

TLDR: A list of quotes that support the different arguments for Young Griff's identity. No real conclusion was meant to be reached.