//I am looking for something different for breakfast.//
Try oatmeal! Or sushi!
Kidding aside, I don't know of ANY place downtown with a drive thru, and most of the haunts I frequent open at 7am. Your best bet is probably Golden Pride on Lomas between Yale and University.
Golden Pride on Lomas by UNMH fits the bill. WE HAVE A WINNER!
What about that Taqueria Mexico joint on Lomas just east of the courthouses? I know they open early. You may actually have to get out of your car and order at the window, though. I think they open at 7 a.m. so I'd think they have breakfast burritos, no?
Here's a link to their menu...
I am not allowed to ride yet, at least not with adult supervision.
Again no drive thru and not the right time, but the cafe in the basement of Plaza del Sol has $2 breakfast burritos- they open at 7.
There is a good burrito truck- Pink Ladies, that had been parking on the north side of the 4th street mall, not sure when they open.