TL;DR version: The Night’s King is interested in Jon Snow because he sees some of himself in the current Lord Commander.
“The gathering gloom put Bran in mind of another of Old Nan’s stories, the tale of Night’s King. He had been the thirteenth man to lead the Night’s Watch, she said; a warrior who knew no fear. ‘And that was the fault in him,’ she would add, ‘for all men must know fear.’ A woman was his downfall; a woman glimpsed from atop the Wall, with skin as white as the moon and eyes like blue stars. Fearing nothing, he chased her and caught her and loved her, though her skin was cold as ice, and when he gave his seed to her he gave his soul as well.”
The Night’s King, rumored to be the 13th Lord Commander of the Night’s Watch, took a particular interest in Jon Snow when the latter shattered a White Walker with his Valyrian steel sword, Longclaw. In Jon Snow he saw a formidable foe.
Here’s why Jon Snow might be of more interest:
Perhaps the Night’s King saw a little bit of himself in the 998th Lord Commander?
One rumor maintains that the Night’s King was a Stark prior to his rise within the Night’s Watch and subsequent fall to the “dark” side.
And what brought him down? A woman from north of the wall. We are led to believe this woman was a White Walker herself, but there is more than one parallel between the Night’s King and Jon Snow.
Both Starks. Or maybe not (for both of them). Both rose to Lord Commander. Both were stricken by a woman north of the wall, which led to some troubles (some of which were greater than others, of course).
It’s takes quite a bit of imagination to think that the Night’s King would know all of this about Jon Snow, but perhaps there’s a magical connection — perhaps he’s drawn to Jon Snow but can’t explain why quite yet.