here is a copy of a post i wrote for someone else but it is most likely your solution as well.
the problem is most likely the logic board solder. this is a very very very popular problem and apple extended a service fix for this well beyond the normal time of apple care but that time for apple extension has past as well.
basically your problem is that the solder under the logic board was not done properly and you will get an occasional screen and then it will go out when you move it or come on when you move it.
the cost to actually replace this is more than the mac is worth and i spent a long time trying to figure out what to do with mine because i wanted it to work but didnt want to put any money into it.
so....
i realize this is going to sound a little crazy but it worked for me and i also fixed another friends this way as well.
my thoughts before i tell you what i did is that i didnt want to spend to the money it wasnt worth to actually replace the problem and i have another mac so if it worked great and if it didnt well then that sucks but i gave it a shot.
1.you basically need to take apart the ibook which you can get the easy to follow directions
here. follow this until frame 30 because you are not going to need to take the top off or at least i didnt.
2. you will be looking for the problem chip on the underneath of the ibook
it is the chip covered with the yellow foam. you will need to scrape off the yellow foam and plastic around it and you will see the ATI card
3. that card is problem and if you exam it you can see that it is soldered to the board from the underside of it. that solder has gone bad
4. ok now we are getting to the fun part and if you read around the web you will here different solutions ranging from sending it off (we discussed the price problem already) to actually using alcohol and starting a fire on top of it inside something like a votive candle. you can actually watch a video of that
here but i was not brave enough to try that for many reasons including not having a fire extinguisher handy
5. make sure you have the keyboard and memory removed as well as the airport card if yo have one.
6. open the ibook up at the end of a table so that the bottom with the problem chip is facing up and the screen is off the table and no longer under the ibook. (hope that makes sense)
7. wrap the entire ibook that is on the table in aluminum foil with 5 layers of foil cutting out the are to only expose the ATI chip
8. go to your local cheapy store like big lots and buy a $10 heat gun
9. put the heat gun on low about 12 inches away from the ATI card and slowly start to bring it closer to the ATI card in a circular motion so that you are sure to get the heat to go under the chip as well as hitting the top of it.
10. i was at first very nervous about how much heat it could take and i actually started with not much and it worked and i put it back together only to find it eventually went bad again and i had to do it all over again.
11. so that being said the lat time i had to do this i started 12 inches away and slowly over like 5 minutes brought it to within almost touching distance of the chip while going around it in circles being sure to get the solder underneath it and the top and i would hang out with the gun this close hwhile doing circles etc... for about :30 secs and then come up to about an inch or 2 away and then back down again and i did that over and over for about 5 minutes.
12. it is going to get really hot so dont panic, just remember to start far away and come down slowly and then when you have completed the whole process above take your time bringing the gun away allowing it to have the heat removed slowly from the chip.
13. DO NOT touch anything or move it for at least 45 minutes allowing it to cool.
14. you can plug it in and test that it worked and it fires up and if it does great put it back together and if it doesnt try again.
i just wanted to add that when i got around to doing this the second time i didnt hesitate to heat it up like i mentioned above until it almost scared me and it worked the first time.
i did this to my ibook more than 6 months ago and it has worked perfect ever since and so has my friends.
hope that helps
good luck
