I've had this with my first iBook a few years ago, and basically the graphics chip has come unsoldered in places, so, after a lot of research I put a shim inside the case to put pressure on the chip and hold the connections in place.

http://www.powerbook-fr.com/ibook/bricolage/repair_g3_video_en_article797.html gives an example, although I used a small coin. iFixIt has more detailed take apart instructions - http://www.ifixit.com/Device/iBook_G3_12%22.

However, it made the case bulge a little on the underside, and as far as I remember didn't last more than a couple of months.

A more extreme solution is heat - http://www.geektechnique.org/projectlab/726/diy-obsolete-ibook-logic-board-repair.html but I really didn't want to take it this far.

A proper repair would mean taking the logic board to a professional and having the solder reflowed on the graphics chip, probably not a cheap option, but more permanent.

Hope this helps,

Cheers

Hugh