Dallas and Omaha are about 588 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 8 hr 35 min on the Omaha 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, Kansas City and Memphis are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between Dallas and Omaha.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.