Hi, Christine. Welcome to Apple Discussions.
I see that my model iBook is a lemon.
That's a bit harsh, don't you think? A "lemon" wouldn't have worked this long for you. While some G3 Dual USB iBooks have had this problem, many more have not.
Has anyone here ever convinced them to fix this model out of warranty?
Yes, they are sometimes making an exception to the three-year rule of the
iBook Logic Board Repair Extension Program, but, being mere users here, ourselves, we don't know what criteria are used to determine which iBooks qualify for the exception to the rule.
It would be my suggestion to try calling Apple again at (800) 767-2775 and ask for
Customer Relations (NOT Customer Service--they can't help you) and ask them to please make an exception to the three-year limit in your case. Polite disappointment seems to get you further than angry demands.
If the answer from Apple is absolutely, positively, "NO!" and squeezing the case to the left of the trackpad seems to help, you may want to consider trying the shim fix suggested by John Sawyer in the Apple Discussions Displays Forum:
[Y]ou can often fix this yourself by removing the iBook's bottom housing, and placing a shim of any sort, about 1mm to 1.5mm thick, onto the raised square on the bottom shield. I use a Scotch mounting square--you can get them in hardware stores and many grocery stores. When you reinstall the bottom case, it will press against this shim, which will press against the graphics chip, and may allow the chip to come into better contact with the logic board.
John Sawyer
CJS Macintosh Repair
Instructions for removing the lower case:
iFixit.com
If Apple doesn't come through, and the shim fix doesn't work (or you're leery of trying it),
DT & T Computer Services in Fremont, California offers the cheapest repair I've found. They have a six-month warranty on the repair.
Once you get it fixed, try never to pick it up by the left front corner alone. Rumor has it that can contribute to logic board failure when the case flexes, causing the graphics chip to come loose from the logic board. If you're picking it up in your left hand (as I often do mine), pick it up in the center under the trackpad.
Good luck.