I actually did replace the battery of a zelda cartridge on GameBoy when I was a kid. The battery wasn't even the right size and barely fitted inside it, so I had to use some tape to hold it together.

That day my older brother treated me like a genius. (That day only)

Unfortunately that trend didn't continue with the GBA, whose games tended to use semi-volatile memory chips that will die after enough time or read/writes. Many of my old GBA games won't hold a save longer than a day or two anymore, without any viable way to repair them. Too bad, so sad.

But hey on the bright side emulators for GBA and GBC are seemingly perfect and run on anything. If someone makes a Gameboy advance micro clone for emulators that is decent I would buy that in a heartbeat.

If you already have a GBA Micro, or other GBA, you can just get a Everdrive GBA: http://krikzz.com/store/home/42-everdrive-gba-x5.html

IIRC many of them use SPI flash or EEPROM, which you should just be able to pop off and replace with a similar part.