The Origin: Multiple places, but a rather fleshed-out version appears on Daily Dot
The Basic Idea: Some believe The Night's King is is directly connected to the Stark line and used to be Lord Commander of the Night's Watch. The logic follows that Jon Snow might suffer the same fate.
The Problem: Well, Jon Snow has not been resurrected as a White Walker, and he's very reluctant to take any sort of leadership role, let alone that of the White Walker King. Plus, he's headed over to Winterfell to take back his home. As he retreats from The Wall, his chances of running into The Night's King any time soon are slim. If you consider the Night's King's legend, which states that he fell in love with an Other, or a White Walker, it seems like a pretty far-fetched path for Jon Snow.