Miami and Raleigh are about 698 miles apart by road. Split the difference and you arrive near Orlando, the city closest to the halfway point between them. That puts roughly 4 hr 18 min of driving on the Miami side and 11 hr 1 min on the Raleigh side — the fairest single meeting point among the cities near the middle.
That's a half-day drive from each side, so Orlando suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Orlando doesn't have what you're after, Tampa and Charlotte are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Miami and Raleigh.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.