New Orleans and Seattle are about 2,099 miles apart by road. Split the difference and you arrive near Denver, the city closest to the halfway point between them. That puts roughly 22 hr 39 min of driving on the New Orleans side and 21 hr 23 min on the Seattle 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 New Orleans and Seattle to converge, splitting the travel instead of asking one side to cross the country.
If Denver doesn't have what you're after, Albuquerque and Salt Lake City are also close to the midpoint and worth a look — each keeps the drive reasonably balanced between New Orleans and Seattle.
City descriptions adapted from Wikipedia (CC BY-SA); photos via Wikimedia Commons, credited above.