Wow, it looks like there are some pretty knowledgeable replies to your issue here. All of the commonly associated issues have been addressed. That's great, that's why you should look here first.
One thing that you may not have tried, although you have had this poor thing in pieces more than together, and have likely either resolved or trashed the machine in the 2 years past. I had a similar issue with an iBook that I still have around here somewhere. Performed all of the recommended and some "don't try this at home" things too.
The solution, was neither obvious nor logical. Often Mac laptops, although super, have some issues with very tight space. One step that I took during my diagnosis and general parts swapping was to replace the bundle cable to the inverter, not the power side, the data side. In doing so I found the bundle to be in an awkward spot by the left hinge and slightly pinched. I replaced this wire bundle and found that the issue went away for some time.
Inevitably to start acting up again in a few months. My solution was to take a small piece of shrink tubing, heated it to slightly bind the wire bundle. Just long enough to get it through the hinge area. Then re-installed the cable in a slightly different location (I believe, just the other side of the hinge), and pushed a little further into the body end of the cable allowing for a bit more tension in the display end... wha-la! No more blinky display or odd shut-downs. It has been almost 1 year and sill going strong, with my niece at the helm (of course).
Since then, when I have seen this type of behavior I will immediately examine the wire bundle on the data side. Since pinching these very thin wires will change continuity of the wire itself, and while not causing the device to fail may cause some very odd behavior. Similar to issues such as "micro-fracture" of the etched surface of the board or "Popped Capacitors", it is much more simple to address and much less costly.
Hope you had good luck with that one, Cheers!