Los Angeles and Miami are about 2,336 miles apart by road. Split the difference and you arrive near San Antonio, the city closest to the halfway point between them. That puts roughly 25 hr 12 min of driving on the Los Angeles side and 24 hr 4 min on the Miami 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. San Antonio still makes a fair, central place for Los Angeles and Miami to converge, splitting the travel instead of asking one side to cross the country.
If San Antonio doesn't have what you're after, Austin and Houston are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Los Angeles and Miami.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.