The only easy suggestion I can make is to clean the game pin contacts with some rubbing alcohol. One of the easier ways to do that is to pick a game, clean it with a good amount of alcohol, so it's rather wet and insert it into the game slot, that'll help clean out the contacts a bit. You could also squish the end of a cotton swab a bit to fit it in there and clean the contacts as well.
If that doesn't help, it's time to open up the Gameboy and look over the PCBs. Luckily they aren't hard to follow and you might be able to find some dodgy sections that are damaged in one form or another.
Try out the rubbing alcohol first and see if that helps. It if doesn't, open up the Gameboy and post some pics of the internals if you can't find any spots that appear questionable. You'll probably need a tri-wing driver to open it up as well.