You gotta love lines like
I'll answer yaAnd the chancellor's got the sense of humor to appreciate the song. That's good since this seems to be a trend, with an Irish "Biddy Martin's Polka" (here), and rumors of a campus band forming that's considering calling itself the Biddy Martins. All of it, as far as I can tell, in a spirit of real affection. Even if you don't agree with her about everything, you gotta like this atmosphere.
I'll dance for ya
We can run this city
But you know Team Verb … always looking for some little language angle. The story, now familiar on campus, is that Carolyn Martin came to be called 'Biddy' from a childhood nickname that stuck. At the same time, there's a little bit of English word history here too, with OED having three separate entries, leaving aside bitty, which is homophonous for most people. The earliest form seems to be the word for 'chicken, fowl', used by Shakespeare, according to the OED. The origins are uncertain, but one possibility is an imitative source (from calling chickens). OED also notes "Gaelic bîdeach "very small'".
The most common use in American English, I suspect, is in collocations like 'old biddy'. This meaning of 'old woman, gossipy', and so on is reported in OED as an Americanism, reckoned as a diminutive of Bridget and used for Irish maids (see also DARE).
From this, it wouldn't be surprising for folks to think of 'Be my Biddy' as negative, but in fact the word has developed a remarkable range of connotations, as UrbanDictionary makes clear, here. It's now used specifically for young girls as well as old women, used for attractive women and women not regarded as attractive, for 'girlfriend' and for women regarded as obnoxious, etc. Even figuring in the necessary (and wide) margin of error for UrbanDictionary data, it's pretty interesting: The word seems to mean both 'X' and 'not X' depending on speaker along at least a couple of dimensions.
There's an old, minor point about patterns of 'semantic derogation', where certain positive or neutral terms develop negative meanings. This has been remarked on especially about terms for women,* where once parallel pairs of terms have diverged, like mister versus mistress or bachelor versus spinster. This might be the reverse case, where a negative word for women has become positive, if in a restricted way. There are all kinds of caveats and limits there obviously, but I'm figuring that some of these uses are genuinely intended as positive.
In the end, in fact, it parallels the situation with coastie, where the word ranges from utterly innocent ('non-Wisconsinite') to, apparently, directly anti-Semitic. Is Zooniversity specializing in lyrics built around lexical-semantic tofu, that is, words that depend heavily on context for their meaning?
*This is may be the classic work on the topic:
Schulz, Muriel. 1975. The semantic derogation of women. Language and Sex: Difference and Dominance. Edited by B. Thorne & N. Henley. Newbury House.