The Original Cloth Dragon: The Sons of the Bright Prince (Spoilers Extended)

Background

Combining the fact that the scope was still smaller in ACoK and the fact that it seems like GRRM had always planned for another Targaryen claimant, I wanted to look at the idea that at one time (and some may argue still) the plan was for the character who ended up being Young Griff to be a descendant of Aerion Brightflame.

If interested: Aerion Brightflame's "ill" act during the Third Blackfyre Rebellion

Introduction to Aerion in ACoK

GRRM gives us the background story on Aerion in Jon's first chapter of ACOK (where like his brothers, "dreams" of dragons killed him):

Aemon took his vows and left the Citadel to serve at some lordling's court . . . until his royal uncle died without issue. The Iron Throne passed to the last of King Daeron's four sons. That was Maekar, Aemon's father. The new king summoned all his sons to court and would have made Aemon part of his councils, but he refused, saying that would usurp the place rightly belonging to the Grand Maester. Instead he served at the keep of his eldest brother, another Daeron. Well, that one died too, leaving only a feeble-witted daughter as heir. Some pox he caught from a whore, I believe. The next brother was Aerion."

"Aerion the Monstrous?" Jon knew that name. "The Prince Who Thought He Was a Dragon" was one of Old Nan's more gruesome tales. His little brother Bran had loved it.

"The very one, though he named himself Aerion Brightflame. One night, in his cups, he drank a jar of wildfire, after telling his friends it would transform him into a dragon, but the gods were kind and it transformed him into a corpse. Not quite a year after, King Maekar died in battle against an outlaw lord." -ACOK, Jon I

as well as the mention of his son (who was passed over for the throne):

Jon was not entirely innocent of the history of the realm; his own maester had seen to that. "That was the year of the Great Council," he said. "The lords passed over Prince Aerion's infant son and Prince Daeron's daughter and gave the crown to Aegon."

"Yes and no. First they offered it, quietly, to Aemon. And quietly he refused. The gods meant for him to serve, not to rule, he told them. He had sworn a vow and would not break it, though the High Septon himself offered to absolve him. Well, no sane man wanted any blood of Aerion's on the throne, and Daeron's girl was a lackwit besides being female, so they had no choice but to turn to Aemon's younger brother—Aegon, the Fifth of His Name. Aegon the Unlikely, they called him, born the fourth son of a fourth son. Aemon knew, and rightly, that if he remained at court those who disliked his brother's rule would seek to use him, so he came to the Wall. -ACOK, Jon I

and if we look at the Hedge Knight, Aerion is a major villain, whereas once we get ASOS and the Sworn Sword, etc. the "villainy" shifts to the Blackfyres. We also have this quote from GRRM in1998 about Aerion:

Aerion Brightfire did not stay in Lys all his life, only a few years. He may have fathered a few bastards there, which would mean Dany has "relatives" of a sort in Lys... but they would be very distant relatives, from the wrong side of the blanket. -SSM, Many Questions: 14 Oct 1998

The Cloth/Mummer's Dragon

Originally I think that this vision in the HOTU that Dany sees was about Maegor (or one of Aerion's bastards) as one of the lies she must slay:

. . . Glowing like sunset, a red sword was raised in the hand of a blue-eyed king who cast no shadow. A cloth dragon swayed on poles amidst a cheering crowd. From a smoking tower, a great stone beast took wing, breathing shadow fire. . . . mother of dragons, slayer of lies . . . -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

which we see in the Hedge Knight somewhat as well regarding Aerion (puppeteer/mummer):

As for the matter of these puppeteers, by the time Aerion is done twisting the tale it will be high treason. The dragon is the sigil of the royal House. To portray one being slain, sawdust blood spilling from its neck . . . well, it was doubtless innocent, but it was far from wise. Aerion calls it a veiled attack on House Targaryen, an incitement to revolt. Maekar will likely agree. -The Hedge Knight

and:

They meant no treason, it was only a wooden dragon, it was never meant to be a royal prince, he wanted to say, but his words had deserted him once and all.

and later with Quaithe's warning:

"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." -ADWD, Daenerys II

Elio's Comments

We also have Elio bringing up Aerion as well:

Elio: the one thing i will say about what we know, and i think i can be vague enough, and i haven't really seen it. i think people haven't thought enough about Aerion Brightflame, and the details of what we learn in the world of ice and fire about him, and how that fits into to things. there's some stuff there that george hasn't really, there's some dots that people have not connected as far as i've seen. so i'll leave you guys with that.

If interested: Aerion Brightflame: Connecting the Dots

Adding of the Blackfyres

Sometime in 1999, GRRM added the Blackfyres to the story which from the information available now Young Griff's background is heavily theorized to be from that female line. From the above information on Aerion, I think that this was either a prophecy/vision shift that GRRM choose to make (similar to the corpse at the prow of the ship) or that there is now some type of "Brightfyre" (great post by u/Mithras_Stoneborne) (combining of the two lineages):

“Dragons old and young, true and false, bright and dark. And you. A small man with a big shadow, snarling in the midst of all.” -ADWD, Tyrion VIII

and:

"My lord does have one prize to offer," Haldon Halfmaester pointed out. "Prince Aegon's hand. A marriage alliance, to bring some great House to our banners."

A bride for our bright prince. -ADWD, The Griffin Reborn

so while I have theorized other fun/unlikely ideas for Aerion's son (the Smiling Knight, member of Egg's kingsguard) I think that there is too much to ignore for it to be something that small.

TLDR: Aerion Brightflame is introduced as a "villain" so early in the story (ACOK, Jon I and the Hedge Knight**), I think it is very possible that the original additional "dragon" (countering Dany, fighting in the Dance of the Dragons II) was supposed to be one of his descendants (either through Maegor or one of his bastards). As the story grew, GRRM added the Blackfyres and potentially chose to combine the lineages or abandon/shift the plotline.**