I had the same problem. It came from me grabbing the computer from the edge. Of course it did not help that the case is just too flimsy to support the weight of the components. I would think that Apple would have thought(tested) of that knowing that most people will pick it up by the edge before carrying it. Therefore, the welds came loose from the graphics chip attached to the logic board...a very common problem with that model.
I just had my logic board reflowed, a solution to the problem, by First Phase Techonlogies in Tempe, Arizona. The cost was $50.00 total cost, which I think is more than reasonable. For that price, they will also return your logic board to you. They have a technician who has been doing this process for over 9 years. Give them a call 1-480-967-1100. Ask for Bobby or Tom.
Of course, you have to remove the board yourself. There are directions on the internet of how to do that if you are so brave. I did it myself and found that it was only slightly painful. You will need a static free mailing bag to send it to them. You can get those from a computer repair shop cheap. The url is as follows:
iBook G3 12" Disassembly: Installing Hard Drive - Removing Top Shield (page 14/17).
TB