Kitty is the best nickname. It's not offensive, you can pretty much say it in public anywhere.
You can yell out Kitty at a crowded shopping mall, don't try that with Pussy....might be some bikers or a cop walking by... end up getting your ass whipped or tased.
Vagina - too textbook
Gash - too surgical
Muff - too cold
Clam - too seafood
Cookie - too Keebler
Cootch - never use this no explanation necessary
Taco - too ethnic
Monkey - too simian...sasquatch
Beaver - too National Geographic
Cunt - too mean, better used as an insult
Snatch - too criminal