This post started off as just a pretty lame summary of the different Snows throughout history and how well the Starks treat them:

He did more than that. The Starks were not like other men. Ned brought his bastard home with him, and called him "son" for all the north to see. When the wars were over at last, and Catelyn rode to Winterfell, Jon and his wet nurse had already taken up residence. -AGOT, Catelyn II

But I think I might have come across something a bit more interesting. In Bran's weirwood dream in ADWD, it could be possible that each vision Bran gets is associated with a "bastard at Winterfell" (not necessarily a Stark bastard).

Bran's Weirwood Dream & Bastards

Note: One problem I have with this theory is that it seems that Bran is going back in time through the visions. If so Lonnel Snow would occur after Dunk visiting Winterfell

Jon Snow

Ned Stark reflecting on his promise to Lyanna and Lyanna playing at Winterfell:

… but then somehow he was back at Winterfell again, in the godswood looking down upon his father. Lord Eddard seemed much younger this time. His hair was brown, with no hint of grey in it, his head bowed. "… let them grow up close as brothers, with only love between them," he prayed, "and let my lady wife find it in her heart to forgive …"

...

The rest of his father's words were drowned out by a sudden clatter of wood on wood. Eddard Stark dissolved, like mist in a morning sun. Now two children danced across the godswood, hooting at one another as they dueled with broken branches. The girl was the older and taller of the two. Arya! Bran thought eagerly, as he watched her leap up onto a rock and cut at the boy. But that couldn't be right. If the girl was Arya, the boy was Bran himself, and he had never worn his hair so long. And Arya never beat me playing swords, the way that girl is beating him. She slashed the boy across his thigh, so hard that his leg went out from under him and he fell into the pool and began to splash and shout. "You be quiet, stupid," the girl said, tossing her own branch aside. "It's just water. Do you want Old Nan to hear and run tell Father?" She knelt and pulled her brother from the pool, but before she got him out again, the two of them were gone.

Lonnel Snow

Lonnel Snow was a bastard son of Wylla Fenn (crannogmen) and Lord Brandon Stark:

After that the glimpses came faster and faster, till Bran was feeling lost and dizzy. He saw no more of his father, nor the girl who looked like Arya, but a woman heavy with child emerged naked and dripping from the black pool, knelt before the tree, and begged the old gods for a son who would avenge her.

Descendant of Dunk

Then there came a brown-haired girl slender as a spear who stood on the tips of her toes to kiss the lips of a young knight as tall as Hodor.

Dunk & Egg are heading north for the next installment in the series.

If interested: What We Know: The She Wolves of Winterfell & Possible Bastards of Ser Duncan the Tall

Brandon Snow

Its possible that Brandon Snow is the youth in this vision

A dark-eyed youth, pale and fierce, sliced three branches off the weirwood and shaped them into arrows.

who later hoped to kill the dragons:

The king's bastard brother Brandon Snow offered to cross the Trident alone under cover of darkness, to slay the dragons whilst they slept. -TWOIAF, The Reign of the Dragons: The Conquest

If interested: The Use of Weirwood as a Weapon in the Series

???

So if this theory holds true this next part of the vision should include something else to do with another bastard at Winterfell:

Then, as he watched, a bearded man forced a captive down onto his knees before the heart tree. A white-haired woman stepped toward them through a drift of dark red leaves, a bronze sickle in her hand.

Then, as he watched, a bearded man forced a captive down onto his knees before the heart tree. A white-haired woman stepped toward them through a drift of dark red leaves, a bronze sickle in her hand.

"No," said Bran, "no, don't," but they could not hear him, no more than his father had. The woman grabbed the captive by the hair, hooked the sickle round his throat, and slashed. And through the mist of centuries the broken boy could only watch as the man's feet drummed against the earth … but as his life flowed out of him in a red tide, Brandon Stark could taste the blood. -ADWD, Bran III

Who could that be? We don't know of any other named bastards that are "at Winterfell" (my extremely subjective definition as I am excluding Ramsay.

It should be noted that (and I can't find the original theory) but I love the theory that the bronze sickle being used was the original "Ice":

Four hundred years old it was, and as sharp as the day it was forged. The name it bore was older still, a legacy from the age of heroes, when the Starks were Kings in the North

So after scouring what we know of northern history, I didn't come up with much outside of the fact that several young Lord Commanders of the Night's Watch were bastard brothers of the King in the North:

"My lord, when I was looking through the annals I came on another boy commander. Four hundred years before the Conquest. Osric Stark was ten when he was chosen, but he served for sixty years. That's four, my lord. You're not even close to being the youngest ever chosen. You're fifth youngest, so far."

"The younger four all being sons, brothers, or bastards of the King in the North. Tell me something useful. Tell me of our enemy."

So the potential exists for a Lord Commander having done something terrible and being executed by the Starks of Winterfell (as we see Ned do to a member of the NW in the first chapter of AGOT).

The only members of the Night's Watch that we potentially know of so far are Osric (mentioned above, ruled for 60 years) and... The Night's King.

Old Nan calls him a Stark here, but the rest fits:

He brought her back to the Nightfort and proclaimed her a queen and himself her king, and with strange sorceries he bound his Sworn Brothers to his will. For thirteen years they had ruled, Night's King and his corpse queen, till finally the Stark of Winterfell and Joramun of the wildlings had joined to free the Watch from bondage. After his fall, when it was found he had been sacrificing to the Others, all records of Night's King had been destroyed, his very name forbidden.

"Some say he was a Bolton," Old Nan would always end. "Some say a Magnar out of Skagos, some say Umber, Flint, or Norrey. Some would have you think he was a Woodfoot, from them who ruled Bear Island before the ironmen came. He never was. He was a Stark, the brother of the man who brought him down." She always pinched Bran on the nose then, he would never forget it. "He was a Stark of Winterfell, and who can say? Mayhaps his name was Brandon. Mayhaps he slept in this very bed in this very room." -ASOS, Bran IV

I think that last part is kinda a stretch, but I liked how much it all lined up "bastard-wise". We will likely find out more about that last vision in TWOW. Its more likely some form of ritual sacrifice to the old gods. And as I mentioned earlier, I wouldn't be surprised if we meet Lonny Snow in the She Wolves of Winterfell, but his birth occurring after Dunk's visit doesn't line up timeline wise.

TLDR: Its possible that Bran's weirwood vision in ADWD, seems to be associated with numerous "bastards of/at Winterfell".