Pudding tame

[Douglas G. Wilson]

Of course in researching the history of "poontang" I came upon remarks to
the effect that this word seems to be reflected in a children's rhyme
(still current, I think) along the lines of

What's your name?
Pudding tame.
[Ask me again and I'll tell you the same.]

In fact "pudding tame" and variants (pudding/puddin' [and] tame/tane/tang)
are used today with the sense "I won't tell you my name" (e.g., often as a
'handle' or pen-name on the Internet, = "Anonymous"). The expression was
used in the "X-files" TV program in 1999.

The rhyme appeared in the US by 1895, when it was cited in "Dialect Notes".
Already we're out of the "poontang" milieu, I think; but in case there's
any doubt, I find quoted from 1861 a version supposedly from ca. 1825
(apparently from Sussex?):

What's yer naüm?
Pudding and taüm.

Back a little further (ca. 1590), I find reason to believe there was
approximately:

[What is your name?]
Pudding of Thame.

Now at least the expression has some surface sense, maybe. Thame is a
place-name -- in particular a town in Oxfordshire, I believe. So "pudding
of Thame" might have been the name of a food, perhaps similar (or at least
analogous) to Oxford sausage, say. Still the expression is meaningless in
the context, and I wonder whether (1) it might even earlier have been
something else ("pudding at home"? "Pudding Tom"? "pudding time"?) which
maintained the rhyme in some early or regional pronunciation, and whether
(2) there is some recognizable double-entendre or other joke here in
16th-century (or earlier) English.

Any ideas?

-- Doug Wilson