There's a brace you can buy for tennis elbow. It goes around the arm just below the elbow and tightens with velcro. The black ones are better than the white ones (I don't know the brands), and they are in most drugstores.

RICE -- rest, ice, compression and elevation -- will also help. NSAIDs, like ibuprofen, can help relieve the pain.

This is tendonitis, and will be a recurring problem if you don't treat yourself right. If you must use a mouse, get a trackball -- there are some very ergonomic ones out there now, but try before you buy. I don't use a mouse at work or at home and I've managed to convert many of my staff -- helps keep the medical costs down, too.