Los Angeles and Tucson are about 439 miles apart by road. Split the difference and you arrive near Phoenix, the city closest to the halfway point between them. That puts roughly 7 hr 29 min of driving on the Los Angeles side and 2 hr 39 min on the Tucson side — the fairest single meeting point among the cities near the middle.
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, San Diego and Las Vegas are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Los Angeles and Tucson.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.