What's the best backerboard to use in a bathroom?

Backerboard or durock as it is sometimes called is mainly for tile applications. It will work in this situation, but may be overkill and does not lend to painting. Green board (water resistant drywall), is used mostly as a backer for tub and shower surrounds and the like. This may be your best option if you are just repairing around the shower head, but I would put a tile or water proof covering over it. A piece of what is called milkboard would be best as it is solid plastic, and can be cut to fit. Seal the back of it with silicone or the like so it does not leak, but it only comes in a couple of colors. White and off-white i believe.