How about Phoenix or Tucson?
As far as the routes, they are similar until you get out into the west. I-40 is a bit more spectacular with high desert scenery and it's especially interesting because of it's proximity to the Indian Nations in New Mexico and Arizona. I-10 is more of a low desert route from west Texas on.
If you go that way, you might spend some time at Big Bend National Park in SW Texas. Search on "Terlingua" on this site and you'll find some recommendations for local entertainment in that area. Also along that route, you could spend some time at Carlsbad Caverns and White Sands Natl Monument, or rest for a couple of days in the high pine country around Ruidoso or Silver City (also New Mexico). In the Ruidoso area, you'll also find Lincoln, NM, where Billy the Kid had a few shoot-outs and escapes during the Lincoln County War in the early 1870s. If you'd like to see that story on the screen, check out the "Young Guns" movies or "Chisum" (an older John Wayne picture). The resorts around Phoenix and Tucson would also appreciate you staying with them for a day or two, and our weather is absolutely perfect right now -- sunny days in the 70s and 80s with cool evenings/nights and crystal clear skies.
If you choose the northern (I-40 to I-17) route, your down time could be spent around Santa Fe and Taos, or Sedona, Grand Canyon or Phoenix. Also along this route you'll find Petrified Forest, Meteor Crater, and several ruins of ancient people's towns.
If you haven't driven either route, I'd recommend doing the more northerly one (I like the scenery better), but that's just my personal choice.