There's not a lot that we know on Game of Thrones about the Night King. This is partly because he is mostly made up for the purposes of the television show. Though the books sometimes speak of "The Great Other," nothing like the Night King has been seen in the novels as of yet, and the "Night's King" in A Song of Ice and Fire is a completely different creature from ancient history. This means any Game Of Thrones Night King theory is fair game.
Considering that, it's a bit of a marvel we don't have more theories floating around as to who that fellow is we saw get "made" back in the Dawn Age by the Children, and what his deal is. Earlier this season, some thought he and Bran might be somehow connected, or even the same person, but that's like saying Harry Potter is Voldemort. Just because one can reach out and assess what the other is doing doesn't make them one and the same.
But after this past week's encounter with the Night King's arrival just as Jon Snow and Co capture their wight, fans are considering a new theory, one that would be Bran and the Night King more alike than we'd want to consider.
What if the Night King is a greenseer?
It's not totally out of the question. During the wars with the First Men during the Dawn Age, The Children of the Forest had human allies, which we assume the man we saw in the flashback might have once been. Those human allies mated enough with the Children that their descendants today carry some of the same traits, like Jojen (greenseeing) and Meera (height and hunting talents.)
It is quite possible that the man the Children turned all those years ago, who has gone on to "make" most of the Army of the Dead who follow him, had that talent. It would certainly explain a few things as well. There's the scene in Season 6 where the Night King is aware of Bran as a greenseer in their reality.
Note that when Bran is visiting other visions, no one seems to notice him, and they only hear him faintly. (Even Wyllis, Young Hodor, who seems to hear him over space and time, only then suddenly is aware of the Bran standing in front of him, and it seems like Bran warging into Hodor in the present while being in the past is responsible.)
That the Night King can reach out and touch Bran, which leads him to find the Three-Eyed Raven, is also something we've not seen anyone else able to do. He also recognizes the ravens Bran's warged into this season.
Then there's this past week, where the Night King seems to have known right where Jon Snow and Co were, and also seems completely unfazed by the arrival of Dany's dragons. Some might suggest that his zombie nature makes him unfazable, but remember the first time Jon struck down a White Walker at Hardhome with his Valyrian steel sword. That got the Night King's attention.
Here it's as if... the Night King knew the dragons were coming.
Could the Night King see the future the same way Bran does? That's seriously a horrible thought. Because it's bad enough that the Night King has an army that only grown the more it kills their enemies, but knowing what's going to happen before you do? That's almost impossible.