I did this on my mouse and it worked out great. Maybe 5 minutes of work.
All you need is soldering iron, solder, solder "sucker" (official name Vacuum Desoldering Tool), and a micro-switch of your choice. Most of this is available at Radio Shack, but for the switch you may be out of luck. Radio Shack carries hardly any components these days. I'm an electronics hobbyist so I already had an assortment of microswitches in my supplies, but DigiKey is a great company for parts.
If your new to soldering and desoldering just practice on a throw away electronics item first. I used to practice removing and replacing the same component on an old circuit board. There are plenty of soldering tutorials on the web...
I don't know if it was necessary but I did sand down my button a bit with medium-grit sand paper. I sanded it down until it was about the same height as the original button. (The plastic plunger part)
As others mentioned you may want to replace both for an even feel. I didn't but I personally don't even notice, I'm just happy I can double-click again.