One of my biggest regrets about not going to the SOR meeting is that I won't be dining in Santa Fe. However, it's been so long that I've actually been there, that I really can't recommend any restaurants. The regional cuisine is what most people think of as typically Mexican: tacos, burritos, etcetera. One ingredient that makes the food "New Mexican" is the Hatch green chile. The chiles are rarely found outside the state; in fact, we see the chiles in Omaha once a year at one of the local supermarkets.
Local restaurants serve the chiles in "green" and "red" sauces. Most often, one sauce is hotter than the other. If you're not used to spicy food, ask your server how hot the food is.
NM Hatch chiles roasting on a grill
Link: NPR's Scott Simon ruminates on the possibility of green chile as a diplomatic tool (from 2003).