I find rain to be very cozy weather, so it's not a bad association at all! Lots of people love the rain, so I don't think you'd really have to worry about "I hate rain" comments.
I'm not really into
Rain as a name, I think I'd be more apt to use
Lorena or
Lorraine nn "
Raine, Rainie"
If I had to use a version of rain, I'd go with
Raine. It feels more namey to me. Rayne is cool, but a bit too edgy for my tastes.