If it's a PCB board that has components on one side and solder runs on the other you can use Isopropyl Alcohol (rubbing alcohol) and some q-tips to remove almost all of the corrosion or gunk that is on the board without ruining anything, honest. Then, after you have it cleaned off you need to examine the connections, solder runs, etc and see if they are still intact. You can then solder in some jumper wires as mentioned, but the caveat is that if you don't know the current running through that board and put in wires that are too thick or thin you could cause some serious damage and even start a fire!
If the car is fairly new, take it to the dealer, tell them you were look at stuff and you found this problem and let them replace it.