Seeing the Bolton banners fall after the Battle of the Bastards on the penultimate episode of Game of Thrones Season 6 was one of the best scenes of the show to date. But besides signaling Jon and Sansa getting back their home, it meant that there has to be a new Lord of Winterfell. But who will rule Winterfell on Game of Thrones now? We all know Littlefinger's obsession with becoming Warden of the North, which is basically the same thing, but there's also Bran to consider. Plus, Jon may be a bastard, but he's been recognized by many of those in the North as someone who has a claim on Winterfell.
But with each of those who have any sort of right to rule Winterfell, there is something holding them back or keeping them from taking place as Lady or Lord of Winterfell. We saw how easy it was for Roose Bolton & Co. to saunter in and take over Winterfell, so obviously the remaining Stark kids (Snow included) aren't going to dip out of their home again. It should also be noted that this is the first time Jon has even stepped foot in Winterfell since leaving for Castle Black way back at the beginning of Season 1 of Game of Thrones.
Catelyn was the one who reminded Ned that "There must always be a Stark in Winterfell", and while there has been, in one way or another, since Season 1, there has not been a Stark ruling over Winterfell since Robb was killed in the Red Wedding. But now that Jon and Sansa have taken back their home, who will rule Winterfell on Game of Thrones? At this point, it could be Bran, the last rightful heir, or any number of these people who, each in their own way, have the ability or claim to do so.
While Jon is still a Snow, or yes, even possibly a Targaryen, he's likely to gain a lot of Northern support now that he's beat Ramsay in the Battle of the Bastards. A legitimate Stark should technically rule Winterfell, but there's a good chance that the important Northern houses won't really care about his legitimacy, and will accept him as the Lord of Winterfell since he was the one who led an army to take it back and won. And also since the only remaining heir by birth right, Bran, is still totally M.I.A. beyond the wall. Also, Bran is basically the new Three-Eyed Raven, so there's that too.
There's this whole terribly sexist thing about ruling a House in Westeros, where you need to be a man or, at the very least, married to a man who has a claim over a country. But Sansa has proven herself time and again to be not only a valuable asset, but a plenty capable strategist. She may not be a man, but she is a Stark by blood and aided in winning back Winterfell from Ramsay Bolton. And at this point, the folks of the North really can't get too picky with their Starks, since there are only a few left.
Petyr Baelish has been after the position of Warden of the North for awhile now and as recent as Season 5 of Game of Thrones has been telling Cersei that with her support, he could rule Winterfell with the Lannisters having control of the North. And now, more than ever, that seems like a real possibility. Littlefinger didn't come in at the last minute with his army of thousands of Knights of the Vale to help defeat Ramsay's army in the Battle of the Bastards for nothing. As he says in the promo for the Season 6 finale of Game of Thrones, and as the letter Sansa sent him promised, he expects something in return. It would surprise no one if that something turns out to be someone. A.k.a., Sansa Stark.
Bran has been gone (again) since Hodor died holding the door on tons of White Walkers, and he was saved by his estranged Uncle Benjen. But with Rickon now dead and Jon still a bastard, Bran is the last true heir of Winterfell. He has Stark blood and carries the name, making him the rightful heir as of right now. How he could possibly rule Winterfell while working double duty as the Three-Eyed Raven is a total mystery, and he is still somewhere beyond the Wall, but yes, Bran is a possible candidate for ruling Winterfell.
Benjen Stark is another Stark who had been missing from Game of Thrones since the very beginning of Season 1, but he made a surprise appearance in Season 6 and was revealed to be some sort of half human-half White Walker hybrid. Could he even go back to Winterfell, beyond the wall? We really don't know much about what he can and can't do, other than swing around a handy mace, but as a Stark, he does have a home in the North and if it came down to it, he's a candidate to rule Winterfell, which he has a claim to. Not that we could really see this non-human as a viable ruler, but he is a Stark.
At this point, the question of who will rule Winterfell on Game of Thrones is anyone's guess, and the logistics of it may not even be ironed out just yet. The Battle of the Bastards is over, but we've still got Dany and Yara's alliance to get the throne back, Cersei's actual trial, and Bran's story (whatever it may be) to get through. The Season 6 finale of Game of Thrones might be full of Battle of the Bastards aftermath, but it's also going to be here to set things up for Season 7. So just slow your roll and let things play out. No matter how hard that is to do.