Does the GBM have that same potentiometer slider based power switch as the other GBA models? Sometimes the inside of the power switch can get a little bit corroded or dirty. I have fixed like five regular GBA's so far that were having power issues by spraying some contact cleaner into the power switch between the gaps in the metal as good as i can, then gently swishing the stick back and forth, then blowing the cleaner out with compressed air while it's still wet to be extra sure i'm flushing residue out completely. The stuff drys out on its own really nice and clean, but I blow it out just to be sure i'm flushing any foreign gunk out of the switch, you don't have to go nuts with the compressed air. Rinse and repeat a few times and it works great. Good for cleaning out the cartridge slot too.
I think I saw wiggy or someone on some other forum mention something like using a clean injection needle filled with isopropyl alcohol to shoot alcohol into the switch and flush it out. That's kinda impractical tho because most people aren't going to have needles like that, lol.
Here's the contact cleaner I use, it's great, but there's other options out there too.
http://www.amazon.com/gp/product/B000BXOGNI