The old school Shimano MTB thumb shifters are a little quieter than the newer rapid-fire types, I think. Many friction shifters are silent, if you want to go back to friction!
Originally Posted by
Retro Grouch
I'm thinking that if everything were perfectly adjusted they wouldn't do that.
If it was my bike, the first thing that I would do would be to check the rear derailleur hanger alignment with a gauge. There probably isn't anything wrong with that, but it's nice to have that possibility out-of-the-way. Then I'd start fiddling with the derailleur adjustments. The Park Tool website has good step-by-step instructions for doing that. Be sure to follow the steps in order because sometimes one adjustment can affect subsequent ones.
On the other hand, your bike is ride able now. Once you start messing with it, you could make it worse.
I think the OP is referring to the clicks the shifters themselves make which there is nothing that can be done (besides gumming them up with sticky old grease.)