I should have seen the clue right away. When I had tried to apply thermal paste to the video chips, I'd noticed that the heat sink didn't quite make contact with the top of the chip. I figured it was because of those pads that were in between, but it bothered me all the same.
Then it started to dawn on me - when I bought this iBook (used) the frame was broken, among other signs of abuse, so I ordered a new one from ifixit.com and transferred all of my innards onto it. First, though, I made a reasonably thorough inspection of the new frame to make sure it was the right one. It looked right, and for a while, anyway, there didn't seem to be any problems.
With a connection beginning to form in my head, I opened up the book again, and made a closer inspection in the area of the heat sink. Lo and behold, I discovered that the machined surface where the heat sink was attached (at the point where the HDD rail screws on) was about 1.5mm higher on the new frame! This was JUST enough to prevent the heat sink from getting a good seat on the video chip. I had only stumbled across it by noticing that the thermal paste didn't get quite squished enough between the surfaces!
So I denuded the frame once again, got out my Moto-tool and milled off the extra metal. I put the iBook back together and booted it up, and twenty minutes later, FREEZE! S**T!! But hey - that's twice as long as it was running before. What else is goin' on?
Once again, I disassembled the machine, and this time, I found that the little stud on the logic board that the heat sink attaches to had been pulled loose, evidently by the upward pressure of the heat sink. I replaced the stud with a screw through from the other side, reassembled my iBook, and folks - it's been running ever since.
So without a positive contact with the heat sink, the top of that chip was overheating and shutting down the video system, and consequently the whole computer. Fortunately, so far there doesn't seem to have been any permanent damage - whew! But why didn't this problem start as soon as I replaced the frame? My guess is that the stud didn't pull away right off, but over time the logic board was allowed to move farther from the heat sink.
A couple of lessons learned:
Be careful where you get your parts! Something like this can happen even with a reliable vendor, so you have to verify everything up front. ifixit, you need to maybe step up the QC, or if there is supposed to be a difference between these frames, we need better information on which one to order.
And, If you lift the heat sink for any reason, when you put it back together, you must make a thorough examination to be sure that there is sufficient contact between the heat sink and all of the chips underneath.
So thanks for all of the responses, hopefully this stuff can help someone else soon.