Birmingham and Washington are about 659 miles apart by road. Split the difference and you arrive near Charlotte, the city closest to the halfway point between them. That puts roughly 7 hr 33 min of driving on the Birmingham side and 6 hr 55 min on the Washington side — the fairest single meeting point among the cities near the middle.
That's a half-day drive from each side, so Charlotte suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Charlotte doesn't have what you're after, Raleigh and Atlanta are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Birmingham and Washington.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.