What do you mean by refurbished?
Any fool with a triwing or was it a triangle for some of the Wavebirds (varies in normal controllers, not sure what different batches have) can take a case off and chuck it in some soapy water in an ultrasonic cleaner, possibly with some cotton bud action somewhere in there if enough disgusting humans have touched it during its life. Wear on paint/dye is a bit harder if you want to go that way, most would say get a vinyl instead if it bothers you so.
Batteries are AA so as long as they have not leaked then is what it is. Some might make some chargeable packs or repair the little sibling damage to a cover but eh.
If they have leaked then that can be harder depending upon what leaked, how far it travelled and what happened. That lands you in general repairs rather than traditional refurb if that is the case.
Sticks on gamecubes usually revolve around people seeking those special ones for smash brothers unlike N64 which grind to dust. Though I have seen a few lose given directions. They are using a plastic that gets a bit gross but that is largely under general cleaning. Replacements can be harder to source and while I don't think the wavebird used different to the just about available wired ones I am not going to say go on ebay and get it from whatever breaker is dealing in them without first checking.
Some modify them in various ways (cstick back to buttons being a fun one) but we can leave that one for now.
I eat my own dog food as it were so have little need of such services, and while there are shops offering repair services for consoles then most are just capable people offering an otherwise basic service if we were to look at the wider electronics world.
