Is there a way to paint or stain a pebble coat pool finish in order to regain or change its color?

Yes there are paints available for pools. I have never heard of a stain for pools though. Pebble surface pools don't "loose" their color. What you may be experiencing is a dull calcium or mineral deposit. Deposits can sometimes be cleaned with an acid wash. If it is very bad it can be blasted off by a tile cleaner.