Back in the day, when I repaired industrial electronics I used to use two products. First I'd go with Tarn-X, the polishing stuff. It is a thin solvent but polishes brass and other circuit board connections. Use a Q-tip. Don't get it on any components, probably won't hurt but I'm not sure.
Second is Flitz, it's a metal polishing compound, use it on a Q-tip and buff the area with it. Get it at police supply or gun shops.
By the way, Hows your 310 doing, I have one that's pretty much blind. It doesn't see any satilites. I'd try a firmware upgrade but haven't found one yet.
Anyway hope I've helped, be careful, don't spill anything on the circuits.
David, in Walker, La.