I've tried a couple things, all depending on how confident you are are in your skills:
I can second the alcohol and eraser. These carts are decades old, a lil' eraser, alcohol, and brasso aren't going to ruin them...
Step one for me if 90% alcohol and qtip doesn't work: Brass Polish (e.g. Brasso). A lot of people will warn against this, simply because it can leave residue. But, I highly recommend this. It will not only get gunk off the pins, it will remove any corrosion from years of air/humidity (kids blowing oreo spit into them) exposure. My suggestion is to remove the board and use something like an "extra hand" to hold the board while you let it dry. You only need the tiniest bit, and don't forget to do both sides. Use alcohol and a qtip afterwards to remove any leftover residue and make sure that it is completely dry before testing. Any moisture leftover can fry the board.
Step two is the above does not work is to locate the memory chip on the boards and refresh the solder. If it is loading a white screen after the gameboy logo it can be because the content after the gameboy screen is failing to load to memory because of a bad joint. These chips are usually the largest on the board, and there may be two, or more. They generally have 16-20 pins (for GB). You really only need enough solder to get the tip of your iron hot and need only to touch each joint to "warm" them up. Sometimes I just swipe the hot iron over the joints from one direction to the other. You don't need to put pressure on the joints or apply significant amounts of solder. I've fortunately only had to do this a handful of times and has worked in all but one (RIP Final Fantasy Legends II). Only on a couple have I had to actually re-solder one of the joints.
Again, only do these if you are comfortable with using brass polish and comfortable with your soldering skills!
Good luck!
Also,
Sometimes on Gameboy cartridges something goes loose on the inside from being dropped on a hard surface in my experience or from careless handling. I mean a loose part in the cartridege itself I think would be the problem sometimes right guys?
Totally agree, had a copy of Starfox that wouldn't load. Found out the tab holding the board in place was broken, so when I would insert the cart in the SNES the board would just float up into the cart instead of seating into the pins. I hot glued a cut up piece of old credit card where the broken piece was and it worked!