I would recommend getting a new battery as Ciprian R said above. If that doesn't work than it might be the actual connecter on the GameBoy. Take a quick peek inside it to see if there are any bent pins or any loose wires or such.
Sounds like a hardware problem, possibly the GPU needs to be replaced. But before you jump to conclusions, I had the same problem on one of my older machines. I would recommend actually pulling out the ram and swapping it out with the other card. To do this will require disassembly. It's not hard, but be careful. Also, a fresh install of the operating system might be in order, but you couldn't see anything if it won't boot.