(n.) a play on words
A caterquibble is a play on words. That’s a word itself that doesn’t have the most robust history behind it, with the Oxford English Dictionary listing just one record of its use in an anonymous poem, Lawyerus Bootatus and Spurratus, or, The Long Vacation, written in 1693.
As seldom used as it may be, though, there’s still a little to be said it.
The element cater has been used to form words bearing some sense of cross purposes since the Middle English period. In the sense of moving in an unexpected direction, or traversing something or somewhere crosswise or diagonally, it derives straightforwardly enough from the French word for ‘four’, quatre.
As for quibble, it started out in the mid 1600s as a nonsense extension of an earlier word, quib, meaning ‘a conversational point that proves evasive’. It derives from a Latin word, quibus (essentially meaning ‘by which?’ or ‘in what way?’), that was apparently once used repeatedly in legal discussions, debates and in-depth conversations to an often confusing and ultimately trivializing extent.
Throw something into conversation simply to muddy an issue or distract people from something else, and you’ve quibbed—and what you’ve thrown into the mix is a quib, or, by extension, a quibble.
Join those two elements together and you have a throwaway comment that seems at cross purposes to its actual meaning or context. In other words, a pun.