While I don't see how old this is, try a new CMOS or BIOS BATTERY (folk call them by different names so let's be gentle on that.)
http://www.hardwareanalysis.com/content/topic/74583/ notes what the battery part number is and another similar story.
It's under 10 bucks most places and any other part such as the mainboard will cost a lot more.
So I've got a Sony Vaio TZ-VGN16GN/B and every time I turn it on, even with the AC plugged in, the buttons flash and the computer acts as if the battery is just dead.
At first I thought it was a bad battery, but even by taking the battery out and using just the AC is doesn't turn on.
It's still possible to turn it on though, I have to plug the AC in and try and press the power button at the same time, which is the weirdest thing I've heard.
I've honestly got no clue, and I really don't want to have to pay much/at all as the netbook was a gift.