Most characters on Game of Thrones have gotten a lot of backstory. Almost too much backstory, if I'm honest. The one exception is the Night King, who is a complete mystery. Naturally, this has lead people wildly to speculate about who he is and how he's connected to the rest of the characters. For example, is the Night King a Stark on Game of Thrones? It's one theory that has been gaining popularity lately.
All that has been revealed about the Night King came from one very particular Bran Stark vision. Thousands of years ago the Night King was just a regular human guy. The Children of the Forest tied him to a tree and stabbed him in the heart with a dragonglass dagger, thus creating the Night King as a means of which to fight against the First Men. They had intended only to create a race that would help them to defeat their human enemies. Instead they got a constantly replenishing army of ice cold death. And that is all that is known about the Night King.
Many fans believe that the Starks are somehow literally tied to the fate of the White Walkers and the North in general. This could be because the Night King was a Stark at one point. It's certainly not the most out-there theory I've heard.
The Starks are said to be descended from the First Men, that is, the first race of men to come to Westeros. This is why they worship the "Old Gods" in their "Godswoods" instead of The Seven like they do down south. The man who became the Night King was definitely one of the First Men, so it's possible he's part of the Starks ancestry.
A popular theory posits that Bran is actually the Night King. The theory speculates that Bran has been traveling back in time trying to defeat the Night King. Some think he first warged into Mad King Aerys and his advice to "burn them all" got disastrously misinterpreted, and then tried again by warging into Bran the Builder, who was the brains behind the creation of The Wall, which was sort of a success. So perhaps Bran tried a final time to change the future by warging into the man about to become the Night King. But what if something went wrong? What if he got stuck there and thereby became the very thing he was trying to stop?
There are definitely holes in this theory and it certainly gives me a headache if I try to think about it for too long. But the theory that the Night King was once a Stark is something I can get behind. Maybe that's why he seems to interested in Bran and Jon Snow. Maybe kin is recognizing kin. Hey, you never know.