I'd recommend either staying in Van Horn if you're coming I-10 or just beyond El Paso at Las Cruces, NM.
You could also take I-10 to Van Horn and pick up 54 north toward Pine Springs. Then west on 62/180 to Hueco Tanks State Park which is very nice for at least a couple night's rest. It has some interesting Indian ruins and is a nice park. Then continue west to I-10 again.
http://tpwd.texas.gov/state-parks/hueco-tanks