Roswell NM

Haystack ohv area has a nice parking/camping area withe covered tables and permanent out houses.

Corp of engineers two rivers dam campground haven't been there for decades. Not much there but tables and nothing.

Bottomless lakes.

Recently saw a video on Bottomless lakes state park. Maybe there?
Bottomless lakes state park

Tatum, NM and Denver City, TX have city parks that are free with hookups. Bit of a drive but if nothing else is closer could work for you.

We were there a few years ago and stayed at Town & Country RV Park...it was nothing fancy, but it was fine for us. It's fairly close to downtown.

The UFO Museum is really neat...I think your son will enjoy!

Alamogordo has a very nice space museum that is worth the visit. Also the US Army base at the west end of White Sands on US-71 has a fantastic display of missles and such tested at the range.

We've overnighted in Tatum. Not a four star park, but heck, it's free and comes with 50 amp full pickups in a safe area. Nothing fancy, but clean and people were nice when we walked into the corner café for breakfast in the morning.

Amazing amount of metal art signs. Little has to win the award for best street signs in all of the United States.

This area has been impacted by the crazy oil rush in southern NM. You will find most RV parks to be full. Best to look for state/county parks where your stay is limited to 2 weeks.

We were disappointed by Roswell, as well as the museum. Took a drive up to the Billy the Kid museum and enjoyed it much better. As smarty stated, rv parks, as well as hotels, are mostly booked with folks working the oil fields.

Try Midway RV park in Dexter, NM. 575-622-0531. We stayed there last September. It is a few miles south of Roswell. Nothing fancy but it worked for us. And the museum was great, allow as much time as possible 'cause there is a lot of info.

2x on the Town and Country. close to town, big sites and the owner is great.
We stayed there for the Alien Festival in 2018. We had a ball.