Chicago and El Paso are about 1,251 miles apart by road. Split the difference and you arrive near Oklahoma City, the city closest to the halfway point between them. That puts roughly 14 hr 31 min of driving on the Chicago side and 12 hr 4 min on the El Paso 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. Oklahoma City still makes a fair, central place for Chicago and El Paso to converge, splitting the travel instead of asking one side to cross the country.
If Oklahoma City doesn't have what you're after, Kansas City and Dallas are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Chicago and El Paso.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.