We claim Andrew Jackson as a native son, despite the preponderance of evidence that he was born in South Carolina (or, as we refer to that state, Baja Carolina).
The Lincoln thing surfaces periodically, but the evidence just isn't there.
We claim to be the birthplace of aviation based on Wilbur and Orville Wright's 1903 flight at Kitty Hawk, so there is at least some truth there -- but the brothers were from Dayton, Ohio, and built their gliders and later their powered aircraft there.
But my favorite is the date proudly emblazoned on our state flag: May 20, 1775. What's that, you ask? Why, it's the date of the Mecklenburg Declaration of Independence, when that backwoods NC county (the locale of Charlotte, which later grew to significant size) declared themselves free and independent of England. Or not. There's not a scintilla of evidence that such an event took place.
Maybe we're just insecure. I dunno. 