The death of Viserion the Dragon in Game of Thrones Season 7 has thrown long terms fans of A Song of Ice and Fire for a loop. All this time, we've been searching for a third person to ride Dany's dragons along with Jon Snow, and probably one who would be a sekrit!Targaryen to boot, and then, no. Nada. It's the Night King, who'll ride an ice dragon instead. But if he's a dragon rider, that leads to fans asking: iIs the Night King a Targaryen?
Honestly, we know so little about the Night King, any and all theories are, at this point, still possible. Part of that is because "The Great Other," as he's called in the novels, hasn't even shown up yet on the page. Everything we know about the Night King on Game of Thrones is what the show has told us. And they certainly haven't said he's not a Targaryen.
What we do know is that most people in Westeros believe dragons can only be ridden by someone who is "blood of the dragon," aka a Targaryen. And our two characters who have an affinity to dragons, Dany and Jon are both Targaryens, even if only one knows it.
So is the Night King, therefore, a Targaryen, since he's now most likely riding one himself?
Probably not. Everything's different now that Viserion is an Ice Dragon. Ice dragons aren't fire dragons. The same rules don't apply here. Where fire dragons were created in Valyria, and semi-tamed by the native peoples there (of which House Targaryen are the last living souls), ice dragons are ridden by ridden by those who have the characteristics of the white walkers.
In George R R Martin's short story The Ice Dragon, the little girl who rides the ice dragon sounds very much like a white walker: blue eyes, cold to the touch, no emotions. It's only after her dragon melts at the end of the story, when his life is over, that she warms back up, and goes back to being a normal child. It seems the only criteria needed to ride one is what the Night King has: being what he is.
Let's also think back to when the Night King was made. It was the Dawn Age of Westeros, a time when the Children were fighting the invasion of the First Men. The First Men would eventually settle down on Westeros after the war and found such houses as Stark in the North, and Lannister in the West.
But the Targaryens are different. They weren't there. They didn't come to Westeros until literally thousands of years later, after the Age of Heroes came and went, after the Age of the Andals was over. The Night King isn't a Targaryen, because there was no such thing.
The dragon's third head is the Night King. But just because all Targaryens are potential dragon riders does not mean that all potential dragon riders are Targaryens. Chances are higher that the Night King is a Stark.