note: to see the Japanese characters properly, you may need to change the language settings on your browser to "Japanese (Shift-JIS)".

The kanji for Lain is: 玲音 .

According to online Japanese dictionaries:

so therefore, Lain = "the sound of jewels"

What that means or what a jewel sounds like, I have no idea.

玲音 can also be written as れね and therefore romanized as "rene", though of course we always hear Lain's name pronounced as "rei-in".