I have come across some posts/comments saying that Jon Snow will stay as Jon Snow and will never become Jon Stark. Here's the thing though - Jon Snow IS already Jon Stark! He just doesn't know it yet.
Q: I have a question, since Robb actually  legitimized Jon and named him his heir for Winterfell and the North  before the Red Wedding (granted no one knows about this and is still  alive or free, the Greatjon knows as does Edmure, but I dont see them  getting out of the Twins any time soon and Catelyn would probably die  before telling anyone) does this make Jon’s rejection of Stannis’ offer  moot?
A: Edmure and the Greatjon are prisoners, true… but you are forgetting  the envoys that Robb sent to Howland Reed… Galbart Glover, Maege  Mormont, Jason Mallister… they are all alive and free. As to what is and is not moot… the key point is, only a king can legitimize a bastard……
"Precedent," she said bitterly. "Yes, Aegon the Fourth legitimized all his bastards on his deathbed. And how much pain, grief, war, and murder grew from that? I know you trust Jon. But can you trust his sons? Or their sons? The Blackfyre pretenders troubled the Targaryens for five generations, until Barristan the Bold slew the last of them on the Stepstones. If you make Jon legitimate, there is no way to turn him bastard again. Should he wed and breed, any sons you may have by Jeyne will never be safe." - Catelyn, ASoS
Robb stood, and as quick as that, her fate was settled. He picked up a sheet of parchment. "One more matter. Lord Balon has left chaos in his wake, we hope. I would not do the same. Yet I have no son as yet, my brothers Bran and Rickon are dead, and my sister is wed to a Lannister. I've thought long and hard about who might follow me. I command you now as my true and loyal lords to fix your seals to this document as witnesses to my decision."
A king indeed, Catelyn thought, defeated. She could only hope that the trap he'd planned for Moat Cailin worked as well as the one in which he'd just caught her. - Catelyn, ASoS
So it is done. King in the North Robb Stark has already legitimized Jon Snow as Jon Stark. Edmure Tully, GreatJon Umber, Maege Mormont, Galbart Glover and Jason Mallister all now know that Jon Snow is Jon Stark.
They just have not managed to either get word to him (Edmure and the GreatJon being prisoners) or for reasons unknown - most probably because of the Boltons squatting in Winterfell with Northmen as Lannister hostages and Stannis engaged in a campaign in the North - have just not informed Jon about this.
This right here is strong foreshadowing for Jon Stark as King in the North through Robb's Will:
Stannis read from the letter. "Bear Island knows no king but the King in the North, whose name is STARK. A girl of ten, you say, and she presumes to scold her lawful king." - Jon I, ADwD
At this point it's clearly implied that Lyanna Mormont has heard from her mother. Maege Mormont - who is witness to Robb's last will and decree which names Jon Stark as KITN - is last seen heading to the Neck with her daughters.
Catelyn wondered if Lady Maege had reached the Neck as yet. She had taken her other daughters with her, but as one of Robb’s battle companions Dacey had chosen to remain by his side. - Catelyn, ACoK
In ADwD, Alysane mentions that two of her sisters are with their mother. Which makes it clear that she has heard from Maege.
"Sisters," Alysane Mormont replied, gruff as ever. "Five, we were. All girls. Lyanna is back on Bear Island. Lyra and Jory are with our mother. Dacey was murdered."
"The Red Wedding." - The King's Prize, ADwD
Which then implies that Lyanna Mormont has heard from her mother and her declaration is for Jon Stark.
Now of course they could be talking about Rickon Stark, if Maege and Galbart Glover managed to communicate with Robett Glover and Manderly has let them know that Rickon is still alive. However, keeping in mind that Galbart Glover was also witness to King Robb's last decree which legitimizes Jon Snow as Jon Stark, it's very possible this is about Jon Stark as well.
The next paragraph just hits us over the head with foreshadowing that more or less confirms that this is about Jon Stark.
Maege Mormont had ridden south with Robb, Jon knew. Her eldest daughter had joined the Young Wolf's host as well. Even if both of them had died, however, Lady Maege had other daughters, some with children of their own. Had they gone with Robb as well? Surely Lady Maege would have left at least one of the older girls behind as castellan. He did not understand why Lyanna should be writing Stannis, and could not help but wonder if the girl's answer might have been different if the letter had been sealed with a direwolf instead of a crowned stag, and signed by Jon Stark, Lord of Winterfell. It is too late for such misgivings. You made your choice. - Jon I, ADwD
Yes Jon, the answer would have been different because Lyanna straight out just declared for YOU! He just doesn't know it yet.
Jon took a knee. The king frowned at him, and rattled the parchment angrily. Rise. Tell me, who is Lyanna Mormont?
One of Lady Maeges daughters, Sire. The youngest. She was named for my lord fathers sister.
The fact that it's LYANNA Mormont making this declaration - which GRRM reminds us of in the same discussion when Jon tells Stannis that she is named after his father's sister - just sweetens the whole deal!