Spanish Harlem
Late to the thread but the folks who commented about there not being good Mexican food in NYC are wrong - it exists in spades in Spanish Harlem!
I can name a place - so I will (here's one: Mesa Mexicana, at 117th & Third - carnitas tacos that are DELICIOUS) - but the truth is I don't need to - the 5 blocks in any direction all around our apartment (we are at 119th & Third) are filled with all kinds of good INEXPENSIVE Mexican food, made by Mexican people. Just walk in this neighborhood particularly on 116th Street and you will run into 10-20 cheap, authentic Mexican spots.
My boyfriend is 1/2 Mexican, raised in L.A., and he is picky about Mexican food having grown up with it. We like Rosa Mexicano and used to go with some regularity but it's expensive and much of the food is not authentic - although some is (guacamole is excellent as many have said, as is the tortilla soup). Dos Caminos is not terribly Mexican at all, if you are going to be picky about authenticity, which I am not (and I admit that I prefer what he considers 'inauthentic'). We've been to almost all of the other Mexican places named and same thing: enjoyable, but upscale, $$$, and not always authentic. Haven't tried La Luche yet.
In sum, Spanish Harlem has what he considers the most authentic Mexican food - and it is DEFINITELY the cheapest. There are panaderias here that he loves as well...with the kind of baked goods that are hard to find for him elsewhere in Manhattan...so if you are on a budget there really isn't a better place to go for Mexican.