To preface: I have a theory about a potential direction Jon Snow’s fate could take in the show. This theory is informed by events in the books, though contains no spoilers for the show from the books.
At this point, the books and the tv show are very, very different, so all bets are off.
If you have not read the books, and want to know nothing about them, then do not read further. Again, this is not a show spoiler, it is a pure theory, but one that would not be possible without the books.
Theory after this photo of Ghost as puppy!
Book info not directly related to anything that has actually happened in the show now: this is the final warning.
So, as we all know, Jon Snow is dead. We all thought Melisandre would resurrect him as Thoros, a fellow priest of the Red God, resurrected Beric after his trial by combat against the best person in Westeros, Sandor Clegane, and about 5 or 6 times prior to that. Melisandre was surprised by Thoros’ ability to do this, so she probably doesn’t know how to do it. She may try to whip up a prayer and entreat R’hllor to save Jon, but that may or may not work...
But we all know Kit Harrington isn’t going anywhere, so how does he come back? Or... as what? We know that corpses return as Wights, but in the book, there is Cold Hands. Cold Hands was a ranger of the Night’s Watch, a black brother in a black cloak, who rides a massive Elk and is surrounded by ravens. He is clearly a wight- dead, frozen, ice blue eyes. But he is also completely sentient and benevolent. Regardless of the theories as to Cold Hands true identity (I do not believe he is Benjen Stark, but that is a separate discussion) we can compare him directly to Jon Snow.
Cold Hands was a brother of the Night’s Watch, and he died and returned. He is also apparently a Warg, as he commands the elk and the ravens and seems to see through them. Jon Snow is a brother of the Night’s Watch, a Stark, a Warg, and he is dead.
Maybe, just maybe, Jon Snow is the equivalent of Cold Hands in the show. He died, and he will return. We’ve all seen that Melisandre has true power, but even she doesn’t know how Thoros saved Beric, so maybe the “obvious” solution will fail. The next step is that Jon comes back as a wight, as any other dead body this far north this close to winter does... but being a Stark, a Warg, and an actor on Contract, maybe he comes back sane and sentient like Cold Hands.