Yes, it can be stained. If you are using the regular stuff with the shiny "skin" on the curved side, it will take some experimenting to get the shade right and it will probably not be an even color.
They make a version specifically for dying that David showed me which has the skin stripped off. It still takes a bit of experimenting to get the shade right but it will be an even color.
I think staining to match the reel seat is a great idea. Staining opens all kinds of possibilities.