Charlotte and Raleigh are about 130 miles apart by road. Split the difference and you arrive near Richmond, the city closest to the halfway point between them. That puts roughly 5 hr 12 min of driving on the Charlotte side and 3 hr 28 min on the Raleigh side — the fairest single meeting point among the cities near the middle.
At this range, Richmond works for a same-day meetup: close enough from both Charlotte and Raleigh to meet for lunch or an afternoon and still be home by evening.
If Richmond doesn't have what you're after, Washington and Atlanta are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Charlotte and Raleigh.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.