Eats in Lubbock
Try Jazz on 19th, just east of the Hospital complex. Cajun and pretty good. Been there a LONG time, which is good for Lubbock.
Completely off the food (Tex-Mex) topic - try Pete's Drive Inns...especially the Pastrami sandwichs. don't go looking for an east coast version, but something completely unique and very good. Pete's are on Ave Q about 10th and on East 34th, East of I-27. Both have been around a LONG time.
Mexican food places come and go quickly.
I lived in Lubbock about 3 years ago, and I have to say, I was quite disappointed with the quality of Tex-Mex out there. However, I came to know and LOVE Vietnamese food by way of a little place called Saigon Cafe on 50th street near Quaker.
If you are looking for something a bit nicer, Stellas (50th near Quaker as well) serves solid Italian food in a more upscale atmosphere.
As far as Mex/TexMex, it was mostly chains, and disappointing chains at that. I'm sure there is something better out there, but I just wasn't in the know.