Birmingham and Los Angeles are about 1,798 miles apart by road. Split the difference and you arrive near Albuquerque, the city closest to the halfway point between them. That puts roughly 23 hr 49 min of driving on the Birmingham side and 13 hr 54 min on the Los Angeles 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. Albuquerque still makes a fair, central place for Birmingham and Los Angeles to converge, splitting the travel instead of asking one side to cross the country.
If Albuquerque doesn't have what you're after, El Paso and Oklahoma City are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Birmingham and Los Angeles.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.