A thought on method. Most of the suggestions here appear to be looking for a "cute" name. While almost any of the examples will attract some, almost any of them will turn others off.
The suggestion would be to use an "ordinary" name that can be associated with something "cute."
I don't kow that it would be a particularly good choice, but an example would be the name "The Old Dun Cow." Inocuous for the public visitor, but when the regulars learn the song - it's mildly rowdy, parodies "commercial" Irish stereotypes - fairly gently, and would be an "inside" joke for the regulars.
The problem is that you have to get at least a few of the regulars to sing it fairly often - at least at first.
Then if the joke runs itself out, you just quit singing it.