Suppose that in the 1850s something changes the sentiment in North Carolina. Basically the people feel that "law and order" is important. This makes them less favorable to rebellion.
When South Carolina leaves the union, secessionist figures in North Carolina seize the federal armories for the state militia. However, to their horror, the convention votes in favor of staying with the union 75% to 10% (plus some abstains). In addition, the unionists outnumber the secessionists 3 to 1 (and I bet most people are "I don't care, just don't loot my farm").
The governor focuses on clamping down of secessionist groups inside his state. He issues a declaration that North Carolina stays with the union, like the convention said, but the state's railroads should not be used for offensive operations against "sister states" (OTL either Kentucky or Maryland made a "we're with the union but don't use our resources against the South, which Lincoln promptly overrode since it's a civil war and you need every mile of rail) and likewise only volunteers can go to the Northern Army (ie, no drafting allowed). He deploys his state militia in anticipation of an attack from Tennessee and South Carolina. The rank and file stay loyal to the state government. As an additional complication Jeb Stuart decides to go unionist and take a role in training northern cavalry units because he thinks his home state of Virginia is doomed to fall to the North even if the rest of the South splits and if it's going to fall anyways, he might as well further his career. He thinks his conscious can be clean if he doesn't actually lead any combat units against his home.
In OTL, North Carolina's state government wanted to be union but felt like it got dragged into the mess. On the lower levels, there were people in both camps. Despite that the common people were more split than their wannabee union government, NC had the highest rate of desertions in the Army of Northern Virginia. Also, unionist resisters there were a pain that the CSA never manage to snuff out, simply chase them away. With the state government firmly in a unionist position and the common sentiment being unionist, the CSA can't establish a rival rebel government.
What happens next? Does Lincoln give NC the Maryland treatment ("we need your rails") or simply wait for the rebels to force NC's hand? If Virginia units face off against Northern Units from DC and Bull run isn't butterflied away (it probably would be, but let's say the war starts off with a Northern fiasco), that leaves Tennessee and South Carolina in a position to invade NC, whose state militia is not very large.