Las Vegas and San Antonio are about 1,071 miles apart by road. Split the difference and you arrive near El Paso, the city closest to the halfway point between them. That puts roughly 12 hr 12 min of driving on the Las Vegas side and 10 hr 31 min on the San Antonio 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. El Paso still makes a fair, central place for Las Vegas and San Antonio to converge, splitting the travel instead of asking one side to cross the country.
If El Paso doesn't have what you're after, Albuquerque and Tucson are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Las Vegas and San Antonio.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.