Birmingham and Phoenix are about 1,452 miles apart by road. Split the difference and you arrive near Dallas, the city closest to the halfway point between them. That puts roughly 12 hr 9 min of driving on the Birmingham side and 18 hr 33 min on the Phoenix side — the fairest single meeting point among the cities near the middle.
Over this distance most people will fly rather than drive the whole way. Dallas still makes a fair, central place for Birmingham and Phoenix to converge, splitting the travel instead of asking one side to cross the country.
If Dallas doesn't have what you're after, Oklahoma City and Austin are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Birmingham and Phoenix.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.