El Paso and Los Angeles are about 700 miles apart by road. Split the difference and you arrive near Phoenix, the city closest to the halfway point between them. That splits the trip almost evenly — about 7 hr 15 min from El Paso and 7 hr 29 min from Los Angeles.
That's a half-day drive from each side, so Phoenix suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Phoenix doesn't have what you're after, Tucson and Las Vegas are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between El Paso and Los Angeles.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.