Boy, that's so familiar. Good question, Kate. I wish I could remember where I used to hear that:

What's your name? Puddin 'n' Tane! Ask me again and I'll tell you the same.

There's quite a discussion of it at wordwizard.com.
I wish I had access to JSTOR. There's an article on the subject in Western Folklore, Vol. 13, No. 2/3 (1954), pp. 190-198 - "Children's Taunts, Teases, and Disrespectful Sayings from Southern California," by Ray B. Browne. This link (click) places it also in New England (and has all sorts of other good stuff).

-Joe-