- From squirrel, evoking the furtive, fearful, scurrying mannerisms of the familiar rodent — the most common theory, by far, and the most likely explanation;
- From Middle English querele (< Old French querele < Latin queri) “complaint, lament(ation)” — going back to same root that gives us querulous — from Quirrell’s whining, complaining personality; or
- From Middle English querele “quarrel, dispute, altercation” — going back to the same root as the previous, but with more belligerent than sniveling connotations.
- And incidentally, it might just be possible that Rowling borrowed the name from Stephen R. Donaldson’s Chronicles of Thomas Covenant, the Unbeliever series, which has a character named Quirrel.
The “perhaps” and question mark indicate that the makers of the OED themselves were not sure of the meaning. A single citation is offered to attest the word, from Richard Stanyhurst’s 1582 translation, The First Foure Bookes of Virgils Æneis: “Soom doe slise owt collops on spits yeet quirilye trembling” (Book I). Of all the strange coincidences, coming across collops again is one of the most unlikely! Stanyhurst’s translation is not well-regarded. No less than C.S. Lewis called it “a monstrosity”, “trounced as it deserves” by most critics, with “no place in the history of even the English hexameter, for it is barely English” . Harsh words!
Whether Rowling had ever come across this word is not at all certain, but it’s possible. She’s admitted to getting names from references like Brewer’s Dictionary of Phrase and Fable (1870) and Nicholas Culpeper’s Complete Herbal (1653), and she’s resurrected a number of obsolete and dialectal words, such as dumbledore, hagrid, and mundungus. Why not the OED?
On the other hand, it seems she isn’t the inveterate dictionary-diver I would have expected. In a 2005 interview, Stephen Fry asked her, “Now do you actually trawl through books of rare words or OED or things, or are they [your names] just things that you somehow, you’ve got a good memory for words?” Rowling replied, “I don’t really trawl books. They tend to be things I’ve collected or stumbled across in general reading.” It seems more than a bit unlikely that quirily would ever come up in general reading. Then again, neither would
 Lewis, C S. English Literature in the Sixteenth Century, Excluding Drama. Oxford: Clarendon Press, 1954, p. 365.