Sorry to have to say:
In my opinion, the rather complex emblem on that button is not in any way related to North Carolina. (Nor is it a Military button.)
There is a name for the geometric shape of the main emblem, but I cannot recall that name. The points can make the emblem kinda-sorta resemble a "star"... but it isn't a star. It is a six-sided figure composed entirely of Arcs. (For anybody who doesn't already know, an Arc is a piece of a circle... the word "arch" is derived from it.)
No button with an even slightly similar emblem is found in the North Carolina section of the Albert button-book, nor in Dan Binder's excellent softbound "companion" booklet, "Civil War Collectors Guide To Albert's Button Book."
I like Mud Hut's suggestion that it resembles Japanese buttons which have a stone mounted in the center. Some "Golden Age" buttons also have a stone (or rhinestone) in the center. All of those were made for Civilian usage, of course.
Micro-examination of the photo of this button's back suggests there may be a backmark which can be deciphered after careful cleaning. If so, that could help pin down the button's time-period of manufacture. Sorry to have to contradict the finder's hope, but IMO this button is far more likely to have been made in the 1840s/50s than during 1861-65. Being found near a civil war button is suggestive of civil war usage... worn by a Confederate soldier to replace an irreplaceable lost Military button on his jacket. The backmark, if there is one, can help settle whether it was made during 1861-65, or not.