Game Of Thrones Season 7: 7 Major Fan Theories About Jon Snow & Daenerys Targaryen
5. He'll Support Dany's Claim
The direct counter theory to the idea of Jon making his own claim, there's the notion that he will indeed support Daenerys' bid. She's summoned him to Dragonstone to bend the knee (although Tyrion did leave that bit out of his message), and maybe that's what he'll do.
It doesn't feel right for the King in the North to bend the knee to someone who doesn't actually rule yet, but Jon is capable of being pragmatic enough to realise that it might well be necessary. And if he's willing to do that, he might be prepared to do more to support her.
Jon is focused on the White Walkers, but if he can be convinced to partake in Westerosi politics, he surely wouldn't want Cersei Lannister sitting on the Iron Throne and, depending on how the meeting goes, would presumably be prepared to back Dany against her. Even if the truth of his parentage were revealed, Daenerys would likely believe she still had the best claim and press on anyway, and Jon - not wanting to rule - may feel he has a duty to support her.
There's also the possibility that this is what it takes for Jon to get what he wants: help against the coming storm. Daenerys has no real knowledge of the White Walkers, and right now no reason to believe anything about them. Jon needs something from her, so she's going to want something in return: support her claim to the Iron Throne, and he could get, at the least, a lot of dragonglass, and possibly some dragon assistance when the time comes too.