The North remembers… SPOILERS!
Before we jump into the possibility of the Starks being connected to the White Walkers, we’ll need to spend a little time talking about the Starks who came before Lord Eddard. Who were they? Who established House Stark, and what is its purpose in the grand scheme of things? The short answer is: they were probably the first good guys to make peace with the Children of the Forest. Aaaand and they were probably the first White Walkers. SHUT.UP. you say? When we found out, we said that, too.
First, we’ll look at the Stark family history in Game of Thrones, their part in ending the Long Night thousands of years ago, and the possibility that their bloodline carries both White Walker and Children of the Forest DNA. No one has George R.R. Martin’s number in their phone, so a lot of the information in this list is patched together from the books, the show, and a few obscure, tinfoil theories floating out there in the GoT fandom ether. It's worth noting, though, that more than a few Game of Thrones conspiracy theories that people thought were crazy-town were eventually proven accurate. Besides, it’s always fun to throw a crazy fan theory against the Wall and see if it sticks.
Who are the Starks, really? If a select few Targaryens have had magical powers (such as Dany), it stands to reason that the Starks could have their own special brand of magic. At least some of them seem to be able to resurrect themselves (though R'hllor may deserve all the credit for that). A few Starks definitely have warging powers, and Bran not only has the greensight, he’s also now the Three-Eyed Raven!
Where does their magic come from? The Old Gods? Something the Starks brought with them when their ancestors came over from Essos on the Arm of Dorne? Neither the books nor the series have outright answered the question, are the Starks Children of the Forest? Or, was a Stark the first White Walker? But there’s a connection. There has to be, right?!
Their sworn houses are Reed, Glover, Mormont (thanks for the 62 men, Lyanna Mormont), Umber (maybe?), Manderly, Flint, Wull, Norey, Liddle, Tully, and Westerling. They have always had trouble with the Boltons, even after House Bolton pledged fealty around 1,000 years ago. As it happens, they were just faking.
House Stark’s sigil is a gray direwolf on a field of white. The house’s words are “Winter Is Coming,” and the ancestral sword was Ice, made of Valyrian steel, now reforged as two swords, Widow’s Wail and Oathkeeper.
There weren’t that many Children to begin with, so the original inhabitants of Westeros did a rash and desperate thing, using an enchanted obsidian dagger—they sacrificed a First Man at a weirwood tree. His eyes turned ice blue. He became the first White Walker.
Here's the theory: the first White Walker was a Stark with the ability to do very Stark-y, magical things. When we see Leaf sacrifice the man at the weirwood tree in the show (the Bloodraven’s tree, BTW), Bran isn’t just seeing any old guy. He’s watching one of his ancestor's getting White-Walkered up.
The Children knew the Starks had greensight powers running through their bloodline, so they wanted to harness that power for themselves by creating a White Walker from a Stark. Instead, they created an extremely hard-to-kill ice monster.