Myself and my family have had 19 laptops (both PC and Mac) over the last 6 years and found that the extended warrantee is a crapshoot. Please see my examples below.
My opinion is as follows:
1) If your child is likely to drop, damage, etc. the machine get the extended warranty. If they are responsible and will be careful with the machine, skip it.
2) If you child is going to be in a location where they can't get service locally in the event of a problem, buy an extended warranty from someone like IBM or Dell. If your child has available backup options (like the Library or home) or a local service place (any CompUSA for example) skip it.
Examples:
I bought a Compaq Presario and was offered a $250 2 year replacement plan on top of the 1 year Compaq warranty. My Compaq had 3 issues within the first year, which were covered by warranty. The last 1.5 years I have had no coverable problems. The battery died, but that wouldn't have been covered anyway. So I saved $250.
My father bought a Toshiba laptop and was offered a $200 2 year replacement plan. The machine had a modem door that held the phone cord in place (poor design by Toshiba), and after 3 years the door lost a hinge. After 3.5 years the springs in the screen hinges (that hold the screen open) died. The modem door hinge cost $50, and the screen hinges were like $200 plus service charges. The 3-year warranty wouldn't have covered the $375 he spent to repair the machine. So he saved $200.
My mother bought a Toshiba laptop and was offered a $350 3 year replacement plan. After 13 months, she was traveling with it and rattled the connector that sends signals from the video card to the screen loose (so the screen didn't have any picture) and cracked the casing. She had to send it in for service from a local place for a total cost of $200. She still saved $200.
My wife was bought a Mac ibook for college and offered the $225 Applecare package. The machine has never had a problem in 4 years (in fact it is the in the best condition of any 4 year old laptop we have ever owned. So they saved $225.
A friend of mine bought a Dell laptop and the 3-year service package. He travels a lot and often falls asleep on the bed with the laptop on his chest. Frequently he wakes up in the morning with the laptop open, on and on the floor. Often the laptop was being damaged during these "transfers" from bed to floor. All damage was covered by the 3 year package. This was a great/smart purchase for him as he save thousands in repair charges after the 1st year (he went through 3-4 screens).