The closest place to halfway would be Fort Stockton. The problem you're going to have is trying to find hotel rooms anywhere near I-10 reasonably close to Midland/Odessa, as there is a true oil & gas boom going on in the Permian Basin, and workers will fill virtually all empty rooms.
Consider instead: 1) heading south onto US 90, where rooms are more likely to be available. You could then stop in Alpine; or 2) heading south on TX 118 and stopping in Fort Davis, which has fewer hotels than Alpine, but is still more likely to have more rooms than Fort Stockton or any place along I-10. Either way, I'd strongly suggest making reservations ahead of time. FWIW, Alpine is a college town, while Fort Davis actually has some tourist attractions. You can continue on US 90 all the way into San Antonio, and it's actually shorter than I-10, although it takes a few minutes longer.
Safe travels!Edited: 08 June 2018, 02:34
As Gene says, do not expect to be able to walk into a hotel and find a room without an advance reservation. Look online and see what vacancies you can find. You might get lucky and find a room in Van Horn or Fort Stockton that's available for a decent price. (We stayed at the best Western in Fort Stockton on our Big Bend trip in November.)
Dropping down to 90 is definitely an alternative, but be advised there are long stretches on that road with nothing at all, including gas stations. If you don't stop in Alpine, you will likely have to go all the way to Del Rio. Marathon only has the Gage Hotel (lovely, historic property, but $$$), there is just one little motel in Sanderson, and nothing in Langtry. Del Rio has plenty of chain hotels and a couple of nice local properties, too, but it's a long way from Las Cruces.
Just to be clear, I had specifically recommended Alpine and Fort Davis because they are roughly halfway. Las Cruces to Alpine is 266 miles, or roughly 42% of the total miles taking US 90 to San Antonio; stopping in Fort Davis is 246 miles, or 39% of the way. (By the way, googlemaps.com suggests turning right off I-10 at US 90 either way; then, after passing the town of Valentine [where the Prada "store" is located*], the next town is Quebec, where you'll turn left [east] and head on TX 166 into Ft. Davis--it appears to be roughly 25-30 miles). If going to Alpine, stay on US 90 instead of going through Fort Davis. (The remaining distance to San Antonio from Alpine is 360 miles, and from Fort Davis is 384 miles. I'd recommend lunch in Del Rio, either way.)
*-The Prada "store" is really not a store, but a work of art in the middle of nowhere. It appears to be a store in the middle of nowhere, and is a great photo op.