Some extra material building on Spagirl’s answer.
Eliot’s godson Tom Faber was the son of Geoffrey Faber, the co-founder of the publishers Faber and Faber for whom Eliot worked as an editor. Tom’s widow Elizabeth put up Eliot’s letters to her husband for auction in 2005. This collection of typed letters includes the first known appearance of “Jellicle” and “Pollicle”:
I Believe that your are to have a Birthday soon, and I think that you will then be Four Years Old (I am not Clever at Arithmetic) but that is a Great Age, so I thought we might send out this
INVITATION TO ALL POLLICLE DOGS & JELLICLE CATS TO COME TO THE BIRTHDAY OF THOMAS FABER.
Pollicle Dogs and Jellicle Cats!
Come from your Kennels & Houses & Flats;
Pollicle Dogs & Cats, draw near;
Jellicle Cats & Dogs, Appear;
Come with your Ears & your Whiskers & Tails
Over the Mountains & Valleys of Wales.
This is your ONLY CHANCE THIS YEAR,
Your ONLY CHANCE to—what do you spose?—
Brush Up your Coats and Turn out your Toes,
And come with Hop & a Skip & a Dance—
Because, for this year, it’s your ONLY CHANCE
To come with your Whiskers & Tails & Hair on
Ty Glyn Aeron
Because your are INVITED to Come
With a a Flute & a Fife & a Fiddle & Drum,
With a Fiddle, a Fife, & a Drum & a Taber*
To the Birthday Party of
THOMAS ERLE FABER!
* (A Musicle Instrument that makes a Joyful Noise)
Oh But P.S. we mustn’t send out this Invitation after All, Because, if ALL the Pollicle Dogs & Jellicle Cats came (and of course they all would come) then all the roads would be blocked up, and what’s more, they would track Muddy Feet into the House, and your Mother wouldn’t Like that at All, and what’s More Still, you would have to give them All a Piece of your Birthday Cake, and there would be so Many that there wouldn’t be any Cake left for you, and that would be Dreadful, so we won’t send out this Invitation,
so no more for the present from your
Silly Uncle Tom.
Tŷ Glyn Aeron, now the TyGlyn Hotel, was the Fabers’ holiday home in Ciliau Aeron in Ceredigion. The ungrammatical “your are” appears twice; if deliberate then perhaps it represents another idiosyncracy of young Tom’s speech.
None of the other versions of the origin story are remotely as plausible, relying as they do on unsupported hearsay rather than documentary evidence:
Eliot heard this word from his young niece, who sounded as if she were saying “Jellicle cat” whenever she called for her “dear little cat” and “Pollicle dog” whenever she called for her “poor little puppy”.
L. Goodman (1991). Letter in Playbill: The National Theatre Magazine 9:7 (30 April 1991).
(Goodman’s letter is rather poorly phrased: as written it says that Eliot’s niece mis-pronounced “puppy” so badly that it sounded like “dog”, which I doubt was the intended meaning.)
According to Andrew Lloyd Webber, when T. S. Eliot was a little boy, his grandma would talk about “dear little cats” and “poor little dogs”—but to his young ears it sounded as if she was saying “jellicle cats” and "pollicle dogs.")
Marty Bell (1993). Broadway Stories: A Backstage Journey Through Musical Theatre. Limelight Editions
A couple of other theories account for only one of the names. Felix Clowder suggested that “Jellicle” is short for “evangelical”, though I am not sure that he was being entirely serious:
Jellicle Cats, it is clear, come out at night to jump about and shout. Does this fact suggest anything? To the casual first glance, perhaps not. But consider: The means adjective means nothing, it is a set of apparently meaningless syllables suggestive only of Edward Lear—until one realizes, consciously or from his own half-conscious memories of small-town tent-revivals, that the author is playing with another and longer word, frequently shortened, at certain social or intellectual levels: E’van-gelical.
Jellicle may thus be either a deliberately sly shortening of Evangelical or
an accurate rendition of middle-class speech.
Felix Clowder (1960). ‘The Bestiary of T. S. Eliot’. Prairie Schooner 34:1, p. 31.
Christopher Ricks and Jim McCue suggested that “Pollicle” originated as a nickname Eliot used to address his friend “Polly” (Doris) Tandy:
TSE addressed letters to Polly Tandy as “Dear Pollicle ma’am” or simply “Dear Pollicle”. At Christmas 1937, he suggested to her a labrador puppy as a possible Christmas present for the children. The epithet came to mean “dog”, as when TSE wrote to Alison Tandy, 20 Dec 1940 about the household where he was lodging for Christmas: “There are also two dogs … There is also a small pollicle which is Australian, and another pollicle is expected by post from Devon.”
Christopher Ricks and Jim McCue, eds. (2015). The Poems of T. S. Eliot Volume II: Practical Cats and Further Verses, p. 43. Faber and Faber.
Based on the dates, the causation surely went the other way, from “Pollicle Dogs” to the nickname. Letters of T. S. Eliot, Volume 8 prints letters addressed to “Pollicle Ma’am” dated 1 December 1937 and 30 December 1937. I cannot find any letter addressed to “Pollicle”, though the British Library has one addressed to “Polligal”, 20 November 1936. This is some years after the appearance of “Pollicle Dogs” in the 1931 letter to Tom Faber.