Historic Old Mesilla has a colorful history that started early with the American Indians and Conquistadors in the early to mid-1800s; it is also the site where Billy the Kid was tried and sentenced to hang on April 13, 1881 (though he ended up escaping later). On the north end of the Historic Plaza sits San Albino Church, one of the oldest churches in the Mesilla Valley. Mesilla is also nice for shopping and dining: La Posta is the restaurant where everyone takes out-of-town guests because of its fun atmosphere; Double Eagle is the fancy restaurant. The Fountain Theater shows artsy movies.
White Sands National Monument is very interesting and unique, great for a hike or picnic. You can rent sleds at the visitor center for sledding down the dunes.
Space Murals, a roadside Americana attraction, features a collection of NASA memorabilia combined with murals by a local painter.
Organ Mountain Desert Peaks National Monument offers lots of great trails at the base of the mountains, ranging from easy (Sierra Vista Trail) to more challenging (Soledad Canyon or Achenbach Canyon).
The Gila Cliff Dwellings are about 2.5 hours from Las Cruces, but well worth the drive. A very scenic hike in a mountain canyon will bring you up to some cliff dwellings that predate the Pueblos of northern New Mexico.