Suggest an optical mouse (no ball to get gunged up). NOT a cordless - these need feeding with batteries or be put on a charger (which many people forget to do - result - mouseless or back to you old one).
I use Microsoft Intellimouse Explorer. Not the cheapest (not the most expensive either, I got this one free though as an extra with M$ Office XP). It's in daily use, all day and still going strong after more than a year. This is the longest time a mouse has worked with me, most previous (cheap) ones giving up after about six months. Just had a look at its underside and there's not a trace of muck or gunge.
I use a piece of white paper as a mousemat which doubles as a doodle pad which I throw out when it's grubby or full of doodles/notes.