My wife and I ate breakfast at this place, last year when we were there.
http://www.mrtimscountrykitchen.com/
While it's not on the River Walk, it is just a short drive if you have a car. I had the Huevos Rancheros.
We also ate at several places on the Riverwalk, including Michelino's and Lone Star Cafe. Both were pretty good.