Mexican, or New Mexican, food is the one cuisine that's quite competitive in Socorro, NM. There are many good options. At this writing, though, my favorite is ten miles south, in the village of San Antonio, NM.
From I-25, take the San Antonio/US-380 exit east (the only way you can go). Turn right on Pino Street, following the signs for Acosta's Bed & Breakfast. This street is one block before you get to the only traffic signal in San Antonio, the blinking yellow light at the Owl Bar & Cafe. The restaurant is a short block south of US-380 on the right.
If the weather is nice you can eat outside. Although the restaurant is small, I've never had to wait for a table. The menu has all the local standards: enchiladas, burritos, rice, beans, and tamales. Everything I've tried here is consistently well-executed. In particular, the tamales are some of the best I've had anywhere. My favorite is the carne adovada, made from pork slow-simmered in a savory red chile sauce. The green chile is just the way I like it: plenty of heat, but not quite life-threatening, and a lot of flavor.
Warning: If you are not accustomed to the heat of our local chiles, ask for the chile to be served on the side, and go gently.