Granite Gap Recreation Area
Near the Arizona border, a spectacular series of limestone and granite ridges mixed with other rock formations rise up to form a secluded bowl around the Granite Gap Recreation Area. This area contains Chihuahuan Desert vegetation at its finest: cactus, ocotillo, agave... And the area is a wintering ground for many different species of migratory birds.
Granite Gap Recreation Area offers no facilities, no restrooms, no drinking water. Hiking and wildlife viewing are allowed but no camping.
To get to Granite Gap: Go to exit 5 on the I-10. From there go south on NM 80 for about 11 miles. There will be an unmarked dirt track on the right (ain't it great?) There's a wire gate at the entrance to the site. Go through it (and leave it the way you found it), take the left fork and go about 1/2 mile along a sandy road. Park near the first large rock cluster, do not drive past the cut in the rocks. Figure on the road being unmaintained and essentially impassable when it's wet.
Granite Gap was open year round and there were no fees involved.
Update, January 2014: I just discovered that the BLM page for Granite Gap is gone. For a while the link was still on the Las Cruces Field Office page but it led to a 404 (File not Found). With the rebuild of the BLM website, all mention of Granite Gap is gone. So I don't know if the property is closed, sold, inaccessible or what.
Maps: BLM - Lordsburg