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 source for this cuisine 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). As you approach the blinking yellow light, you will see two famous burger joints. On your left is the Owl Bar & Cafe. On your right is the Buckhorn: because I don't trust feedlot beef, I've never eaten there (all they have is burgers), but owner Bobby Olguin is nationally famous for having won a round on Throwdown with Bobby Flay.
Just past the Buckhorn on your right, turn right on Pino Street. The San Antonio Crane is directly on your 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.
Of particular merit are the chile rellenos, a whole chile pod stuffed with cheese, battered and fried. There is a pretty wide range of opinion on what to use for the batter. The ones you get at the State Fair are typically wrapped in Krusteaz pancake flour. I personally prefer a light batter of just beaten egg. The rellenos here are exactly right for this style.
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.
For many years, this restaurant was called Acosta's. The current operation is run by Zolla Acosta, daughter of Maria Acosta who ran the original place. The food was terrific then and it's terrific now.