I've been living in Las Cruces for about 4 years, and thought I should finally do a write up on what's good to eat here in case anyone is passing through. Unfortunately, it's a rather short list. Jim Leff's 5 year old summary is pretty much correct: "a veritable chow desert" where a "glimmer of competence is like salvation". (link below)
The upside is that Southern New Mexican cuisine is a distinctive local cuisine that you've probably never experienced unless you've been here before. It's based around chiles (fresh green and dried red), meat (mostly beef and pork), and starch (potatoes and flour tortillas). About the only green things you'll see (other than the ubiquitous chiles) are cilantro, lime, and iceburg lettuce.
The downside of the cuisine is that it's monotonous, and very rarely prepared with any particular love or care. There isn't a lot of subtlety: it burns, it burns! Almost every establishment has an identical menu, and probably purchases their ingredients from the same wholesale provider. Cheese is preshredded, tortillas are store bought, and outside of Lent your chances of finding tasty vegetarian fare are practically nil. And no matter what you order, you'll get the same question: red or green, meaning do you want this smothered in red sauce or green sauce.
This isn't to say that you won't enjoy eating out here on your way through, only that it's the type of food rather than the quality that will be the highlight. Pretty much anywhere you go, they'll have things you can't find anywhere else: stuffed sopa(pillas), gorditas (like sopas but from corn masa), red chile and potato burritos, flat (not rolled) enchiladas, with green chile or red chile sauce smothered on everything. And almost always for reasonable prices: $10 is a pricey entree around here.
The best thing to eat in town is the brisket burrito at Bosa Donuts (190 Avenida de Mesilla at S Main): beef brisket and coleslaw in a flour tortilla. The owner said he got the recipe from someone in Chihuahua, Mexico, so it's not really local but still great. Their donuts are OK, the rest of their burritos aren't as good.
The second best thing to eat in town are the burritos at Go Burger (1008 E Lohman at Espina). Today I had the "Chile Relleno Burrito" and the "Green (pork) and Potatoes Burrito". Their burgers are OK but nothing special, their other New Mexican fare is slightly above average for the town but not great. The combination plate is an accurate sample of the local cuisine. Outdoor seating only, closes early (6:00 I think).
The next best thing in town (and we are already starting to stretch a bit) is the posole at Andale (Avenida de Mesilla in Mesilla). A tasty stew with red chile, pork, and hominy, you doctor it up yourself with lime, onions, and cilantro. Better than the one Jim mentions at Mirador, in my opinion. Their other entrees are fine, but not outstanding. If you are here for the posole, order and enjoy them, but I wouldn't go just for them.
If you want the overall local experience, and are less concerned with the food quality, I'd suggest Roberto's (908 E Amador Ave near Espina). Plastic trays, disposable utensils, very inexpensive, and about as busy at lunchtime as Las Cruces gets. Order anything that catches your eye (say, one red potato sopa and one green meat burrito), or if it seems to complicated just choice one of the combination plates (for example a chile relleno, a flauta, and an enchilada with rice and beans).
If you are looking for a little more upscale (metal utensils and all) Nellies Cafe (1226 W Hadley Ave near Valley Ave) is as good a choice as any. It's a wonderfully friendly place, but I've never really enjoyed their food. But get a combination plate and enjoy the bilingual nature of the conversations around you. They are open for lunch only, and their food is hotter than most.
If you want more upscale, and if you don't mind touristy, La Posta (on the plaza in Mesilla, http://www.laposta-de-mesilla.com) is probably your best choice. Choose any enchilada plate. Their margaritas are pricy but better than most local things.
Oddly, for the best example of the local cuisine, you have to leave Las Cruces and go to the Little Diner in Canutillo, Texas (about 30 minutes south, very hard to find without direction, http://www.littlediner.com/). Same menu as everything in Las Cruces, authentically local and friendly, but the food is prepared with a little more care and attention than the local norm.
People rave about Chopes (Hwy 28 in La Mesa, about 20 min S of Mesilla), but I think that their enjoyment is more due to the pitchers of cheap (in quality and price) margaritas than the quality of the food. It's not that the rellenos are outright bad, it's just that they could be so much better if they used a decent cheese to fill them, fried them at a proper temperature so they weren't as greasy, served them hot, etc. Don't let this discourage you too much: go there and enjoy it, but if you've already eaten a lot of rellenos elsewhere don't expect an epiphany. Regarding an earlier post, I agree it's a dump but I wouldn't be any more hygenically worried about eating there than anywhere else in town.
Pretty much all the other places I've seen mentioned here (Si Senor, Nopalitos, My Brother's Place, El Comedor, etc) are all OK, and while they vary in decor and atmosphere they have pretty much the same food. All decent, none great. Go there and have good time, but don't expect culinary wonders.
There are a couple decent Mexican (instead of New Mexican) places in town. The tacos at Chihuahua's (1750 N Solano Dr near N Main) are particularly good, and they are one of the only places that makes fresh tortillas. They are about the same quality as the Taco Tote chain in El Paso that Jim refers to. Taco Mexicano #2 (1602 S Solano Dr near Missouri) serves good tortas and tacos. Taco Mexicano #1 (1355 N Main St near Picacho) serves about the only seafood in town: their ceviche and cocteles de camarones are quite good.
For other cuisines, there's not much. International Delights has passable Middle Eastern if you don't mind half-cooked falafel. In a pinch, Lorenzo's has edible Italian. Lemon Grass Thai won't hurt you if you stick to the commonly ordered dishes; if you stray from red or green curry your food will likely be prepared by a local high school student who will burn the food to a crisp because he has no idea what the dish is supposed to taste like. All American BBQ (S Valley, next to Walmart) has good smoked brisket and sausage, but unfortunately lousy sauce.
Updates to Jim's article:
Contrary to Jim's guess, the Double Eagle in Mesilla isn't very good, although it isn't clear if this is because they don't know or don't care.
In Deming (1 hr W of Las Cruce) Mirador is better than most options, but isn't great either. The Italian restaurant in Deming is surprisingly good but with awful service. One of the Mexican places downtown isn't bad. And if you are in Deming or passing through, the downtown Deming-Luna Museum (which doesn't serve food) is amazing, and worth a visit if you if you possibly can. Whether you are going for the kitsch (the doll room, the Hispanic Culture room) or the history (the iron lung, the great collection of Mimbres pottery) it will exceed your expectations 10-fold.
In Silver City (1 hr N of Deming) Big Bo y Burritos was indeed fabulous, but has now changed hands several times and when I last checked was no longer in business. As best as I can tell, the rest of the food in Silver City is no real treat.
In El Paso (1 hr S of Las Cruces), Taco Tote, a Juarez based chain, does indeed serve good tacos. Their cheesecake is also quite good, and likely to be different than any other cheesecake you've had: less sweet, fuller textured. There are a few other decent places in El Paso, but this may be mostly in comparison with the lack of places in the rest of the area. I particularly like Pachanga (222 Texas Ave, little or no English spoken). Get the fruit salad with yogurt, honey, and granola and/or a ham, cheese, and pineapple torta.