My mother has stage 3 breast cancer. Could she still battle it?

Yes, curable. Although it can be a difficult treatment with a combination of surgery, chemotherapy and possibly radiation, most of these treatments are aimed at getting rid of the cancer and preventing it from coming back. Unless she has many other or more serious health concerns that make it unlikely for her to live another year or two, then some form of treatment is a good idea.
Yes she can get well. Stage 3 breast cancer has fairly good treatment. If treated properly nearly one half of such folks get well and recover fully and can lead normal lives. So she should not give up hope and get proper treatment and expect to be around for quite a while.