I had always heard it was a protest song about cost of living and politics, and that the weasel was the pocketbook, the monkey being the politician who pursued the citizens' weasel by raising taxes. Opening the pocketbook = pop goes the weasel.

I believe I read an explanation like this long ago, in some erudite tome, but I have not the vaguest idea where. Pop that in your weasel and smoke it ! ">)