I love this series. Jane got the box set a while back, but the tune is very haunting.
The song is called 'Duvet' by a UK band called Boa BTW.
[ame="http://www.youtube.com/watch?v=puARhRd1G2o&feature=related"]YouTube - Serial Experiments Lain 1 (part 1)[/ame]
Edited to include the full beginning including spooky voice and a bit of the cartoon itself.