Chicago and Las Vegas are about 1,521 miles apart by road. Split the difference and you arrive near Denver, the city closest to the halfway point between them. That puts roughly 19 hr 15 min of driving on the Chicago side and 12 hr 41 min on the Las Vegas 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. Denver still makes a fair, central place for Chicago and Las Vegas to converge, splitting the travel instead of asking one side to cross the country.
If Denver doesn't have what you're after, Oklahoma City and Omaha are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Chicago and Las Vegas.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.