It has been well-established that the Starks are descendants of the First Men who found themselves engaged in a battle with the Children of the Forest before the arrival of the Andals. Their history is intrinsically tied to the North, and that could explain why Bran and Jon are both so drawn to the battle against the White Walkers. The Starks are determined to defend their home and the realm, but what if their blood is what provides them with a direct link to the Night King? If he was a Stark when he was human, then that could explain how he could see and touch Bran, and it could possibly also hint at why he seems so focused on Jon. The blood of the First Men could form a bond between the Night King and his present-day descendants that allows them to feed off of the different forms of magic they both seem to possess.
The books offer a small clue that there is indeed a connection between the Starks and the Night King, since the book's version of the character is actually the 13th Lord Commander of the Night's Watch, who is a rumoured ancestor of the Stark family. However, his origin story doesn't involve the Children of the Forest.