A series capacitor is a technique that is used with some frequency to slow certain types of fan motors in the HVAC industry (been there, done that). This usually works well with shaded pole or PSC motors. It will not work on a motor with a separate start winding that is switched in and out with a centrifugal switch or a 3 phase motor. Just monitor motor temperature, it should decrease, not increase with the capacitor value you choose. You can also monitor power consumption with a kill-a-watt or the like to see if you are on the right track.
A series cap has the same end effect as a triac control, it just increases slip. The capacitor will be quieter. I have not yet encountered the resonance issue mentioned by Seekonk, but it is something to be aware of.
As someone else mentioned, the cleanest and best solution would be to try to find a different fan with lower noise specs.