Dallas and Denver are about 663 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 4 hr of driving on the Dallas side and 10 hr 34 min on the Denver side — the fairest single meeting point among the cities near the middle.
That's a half-day drive from each side, so Oklahoma City suits an overnight or a weekend rather than a quick coffee — long enough to want a reason to stay, short enough to drive.
If Oklahoma City doesn't have what you're after, Albuquerque and Kansas City are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Dallas and Denver.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.