There aren't many songs that I like that make me sad because I'm not sad listening to good music. I can be made sad by listening to bad music, but I don't want to put up any more songs I hate than what I need to. So I present to you Tom Waits, A Little Rain.
The lyrics are sad, especially the last verse:
"She was fifteen years old and she'd never seen the ocean, so she climbed into a van with a vagabond. And the last thing she said was I love you, Mom, and a little rain never hurt no one."