Nashville and Philadelphia are about 683 miles apart by road. Split the difference and you arrive near Columbus, the city closest to the halfway point between them. That puts roughly 6 hr 59 min of driving on the Nashville side and 8 hr 42 min on the Philadelphia side — the fairest single meeting point among the cities near the middle.
That's a half-day drive from each side, so Columbus suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Columbus doesn't have what you're after, Pittsburgh and Charlotte are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Nashville and Philadelphia.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.