1. Try resetting the PRAM: http://support.apple.com/kb/ht1379
2. Try resetting the the PMU: http://support.apple.com/kb/HT1431
3. Swap out the Battery and Charger with a friends, or purchase a new set (it's always nice to have a spare set)
4. Check the ram, often the Mac will refuse to start when one or more ram chips is bad: http://www.ifixit.com/Guide/PowerBook+G4+Aluminum+15-Inch+1-1.5+GHz+RAM+Replacem ent/220
5. Maybe try swapping out the hard drive, its worth a shot? Here is the guide:http://www.ifixit.com/Guide/PowerBook+G4+Aluminum+15-Inch+1-1.5+GHz+Hard+Drive+R eplacement/227
In a worst case scenario your Mac may need a new logic board, or graphics card.
Repairs are at your own risk!
Oops just saw your PowerBook is a 12" model. Here are the appropriate links!
Hard Drive Replacement: http://www.ifixit.com/Guide/PowerBook+G4+Aluminum+12-Inch+1-1.5+GHz+Hard+Drive+R eplacement/548
The PMU and PRAM guides are the same for all models!