Game of Thrones Season 6 Discussion [SPOILERS WITHIN]

[Aegon]

Jon's claim is hugely weaker than Dany's.  In the world of Westeros, bastards can't inherit anything, let alone the throne of the entire country (the fact that Jon is even being considered so highly in the North is already extremely unusual).  The only way Jon would have the legal right is if his father legitimized him (like Roose Bolton did for Ramsey), and Jon's father(s) are both dead.  

That being said, Dany could just...force everyone to be okay with it since..y'know, dragons.

But is Jon a bastard? There's a number of things pointing to Rhaegar having legitimized Jon prior to his death. 

The first being, if Rheager had actually just raped Lyanna and fathered a bastard, why would the Kingsguard be there? There's a monumental war happening, why wouldn't the Kingsguard (especially Arthur fookin' Dayne) not be either protecting King Aerys or fighting alongside Rheagar on the battlefield? Why would they be guarding the Tower of Joy unless there was somebody very special in there....like a Targaryen prince, the third head of the dragon, and the Prince That Was Promised? Gerold Hightower, one of the Kingsguard, says in the books that the reason they are there and not protecting Aerys or Rhaegar is because "We swore a vow." That would make no sense unless Jon was a prince, someone Rheagar believed to be the future of the Targaryen lineage in the event that they all die in Robert's Rebellion (which is exactly what happened.) 

Other reasons are fan theories/speculation. The crypts of Winterfell are thought to have something very important in proving Jon's legitimacy. In the novels, Jon constantly dreams of the crypts, of them calling to him. And dreams in ASOIAF very regularly translate to reality. Lyanna is the only woman to have her own tomb in the crypts of Winterfell...hell, the only woman to have a statue anywhere (aside from The Mother). Many readers theorize that Ned did this to keep a relic there, safe from anybody else (especially Robert) that proves Jon is of Targaryen lineage. Many speculate that it's Rheagar's harp that he left behind at the Tower of Joy. 

All that being said, when Jon finds out I don't think he'll actually give a damn. He's lived his life as a Stark (a bastard, yes, but this past season proves that he is truly Stark through and through). His battle is with the North against the White Walkers. He'll have no interest in the throne or the wars of southron lords.