There's a theory floating around on Reddit that Bran Stark is the Night King. Or inhabits the Night King. Check out the Reddit link for more, but the basic theory is that once Bran starts warging back in time, affecting Hodor, then meeting his father at the Tower of Joy - where he finds out about Jon Snow's lineage - he starts going too far.
It even theorizes that Bran tries to alert Aerys Targaryen to the threat of the Night King and drives Aerys mad and he starts stockpiling wildfire and doing the burny-burny thing to people.
Further, as Bran's power grows, he decides that he can find a way to stop the Night King and the Army of the Dead from invading. In doing so, Bran ignores the Three-Eyed Raven, who warns that if you go too far and stay too long, you can never come back. Bran goes back to the last long night and that winter and ends up inhabiting and becoming Bran the Builder, the Stark who constructed the wall and built Winterfell. But Bran doesn't stop the Night King, because, as the Three-Eyed Raven (Max von Sydow edition) tells him, "the ink is dry". Everything that will ever happen is always going to happen. Time is a closed loop.
Ignoring advice, in another attempt, Bran goes back to the creation of the Night King to try to stop his creation by the Children of the Forest. The theory goes that Bran wargs in to the man, tied to the tree, just before the Children shove a dragonglass dagger into his heart, creating the Night King
The theory that Bran is the Night King works, mostly, but there are some flaws. For starters, the Night King rides the reanimated dragon Viserion. And we've been told that only Targaryen's can ride dragons.
So am I suggesting that the Night King a Targaryen?
Yup! I'm doing exactly that.
Follow me: I like the "Bran is/inhabits the Night King idea".
But let's go farther. Look at the guy, a First Man, who takes the dagger in the heart and becomes the Night King. (The Children of the Forest create the Night King to protect them from both the Others - White Walkers in the very upper north - and from men). What color is his hair? Is it... blonde? The scene is sepia-toned, but, it appears to be.
Who has blonde hair? That's right, Targaryens do.
By the way, where is the ancestral home of the Targaryens? Dragonstone. Home of Dragonglass, the only thing, along with Valeryian steel, that can kill the Night King, but the thing that made him as well. Neat symbolism.
Who walks through fire? Targaryens. Dany at Drogo's funeral pyre. Dany when she kills the Dothraki leaders. And the Night King, when he walks right through it, and in to the Three-Eyed Raven's tree cave.
There are a million theories, but tell me why this wouldn't work? Old Nan vaguely tells Bran, as he is laying in bed, after getting shoved out of the window by Jamie, that she claims to know a version of the tale where the Night King was a brother to the King-in-the-North, who was named Bran. Old Nan is suggesting that the Night King is actually a Stark. Does Nan's reference to Bran mean Bran the Builder, maker of the wall and founder of Winterfell? She's vague, and the timeline is unclear, but the Brandon Stark she mentions, even if he's not the Builder, is still a Stark. Could the Night King be a brother to Brandon Stark by marriage?
OR, could have been a brother-in-arms in the Night's Watch sense of "brother" to Bran Stark? What if he is one of the first of the line that would eventually split from The First Men and the Starks into what would become House Targaryen as they moved south after the fracture of the First Men? Was he was there at the very formation of what would become the Night's Watch, and was he captured by Leaf and The Children and turned into the Night King?
Flash forward 8,000 years: could the Night King that our Bran Stark is warging to and (maybe) becoming a part of, the one that rides dragons like a Targaryen, actually BE a Targaryen? Why not? When The Night King touched Bran in his warg, something happened.
Now consider Jon Snow. We now know that Jon is Aegon Targaryen, the sixth of his name. He's also a Stark. Dad was Danerys' brother, mom was Ned Stark's sister.
So if Jon Snow is Targaryen/Stark, and the Night King is/was originally Stark/Targaryen, and Jon's adopted brother Bran is inhabiting the Night King, this could go a lot of places.
Another thing:
Have you seen the spiral symbol of the Night King, which is actually a derivation of the symbol of the Children of the Forest?
I don't know how this all ends. Is there a showdown between two Targaryen/Starks in Jon/Aegon and Bran/Night King? Does Arya have to step in, as she has a Valerian steel dagger - the one she used to kill Littlefinger? Does Arya take a blade for Sansa using her face-swapping ability she learned in the House of Black and White?
I don't know. But the Night King/Targaryen symbols are too close to not mean something in a show where every word means something.