Cincinnati and Cleveland are about 222 miles apart by road. Split the difference and you arrive near Columbus, the city closest to the halfway point between them. That puts roughly 2 hr 30 min of driving on the Cincinnati side and 3 hr 9 min on the Cleveland side — the fairest single meeting point among the cities near the middle.
At this range, Columbus works for a same-day meetup: close enough from both Cincinnati and Cleveland to meet for lunch or an afternoon and still be home by evening.
If Columbus doesn't have what you're after, Detroit and Pittsburgh are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Cincinnati and Cleveland.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.