Lila, according to my name dictionary [Dunkling 1983], is a variant spelling of
Leila where that name is pronounced LIE-la (it is also pronounced LAY-la).
Lila may also be a short form of
Delilah. I suggest you ask this question on the BtN facts board, and refer to this thread as well:
http://www.behindthename.com/bb/arcview.php?id=498497&board=gen
It seems you can choose when you use it, whether it is meant to be based on Lilah (lilac),
Leila,
Leela,
Delilah,
Lyle, or what have you.
- mirfak