The Lincoln NF is vast, and outside of hunting season as deserted as any wilderness around. Much of it will also be under snow right now. Throw a dart at the map and have fun, but the area around Ruidoso/Ski Apache suffered widespread fires recently.
If the snow camping fits your fancy I'd recommend heading to the area around the village of Cloudcroft (8000' ish). The 'Rim Trail' starts just south of town at the turnoff to Weed, and follows the western rim of the Sacremento Mountains for atleast 16 miles, including numerous overlooks of the White Sand N.M., four thousand feet below.
If you prefer more warmth, consider an overnight in the backcountry of White Sands National Monument itself; spectacular with a near full moon! Dogs allowed, but no wood fires.
The drive would probably preclude you from going as far as the Gila N.F./Wilderness on the western side of the state, but if you dare, Turkey Creek hotspring area is classic southern NM backcountry.