In British and Irish English, the jocular phrase thank your mother for the rabbit(s) has been used:
– as an ironic expression of gratitude;
– as a goodbye.

These are the earliest occurrences that I have found, in chronological order:

1-: From the account of a police-court case, published in the Eastbourne Gazette (Eastbourne, Sussex, England) of Wednesday 21^st May 1913:

At the police court on Monday Arthur Greenslade, chauffeur, of Sutton, Surrey, was summoned for assaulting George Ernest Wilton on May 7 […].
[…]
Wilson deposed that he went to the Pier Theatre with Mr. Phillips and Mr. Roberts. […]
[…] Witness and his friends had been to see “A Cigarette Maker’s Romance.” They were not making any noise or suggestive remarks. They were only talking about the voice of a lady on the stage. A lady who was sitting in front asked them to be quiet. One of his friends said, “We will try and forget it.” Defendant spoke to him, but he did not hear what he said, and remarked, “Did you speak or did your ears move?” Defendant struck at him in the theatre. When they were outside the lady said, “You are no gentlemen.” Phillips said, “You are no lady.” She remarked, “Thank you!” and Phillips said, “Don’t mention it. Thank your mother for the rabbits.” (Laughter.)

2-: From the column Tea Table Talk, published in several British newspapers in September and October 1913—for example in The Bicester Herald (Bicester, Oxfordshire, England) of Friday 19^th September:

Lady Tree * insists on trying to make her comrades laugh during the progress of the piece whilst she acts. One night, when she was playing the part of an elderly lady in “Diplomacy” she quite suddenly invented a new line in the play by saying “Thank your mother for the rabbits” to a parting guest. The audience enjoyed it so much that the actress has kept in the line ever since.

(This paragraph was reprinted in the Goulburn Evening Penny Post (Goulburn, New South Wales, Australia) of Saturday 8^th November 1913.)

* The English actress Helen Maud Holt (1863-1937), the wife of the English actor and theatre manager Herbert Beerbohm Tree (1852-1917), was professionally known as Mrs Beerbohm Tree and later as Lady Tree.

3-: From one of the “lists of the ten best known popular catch-phrases that have been in vogue within the last fifty years”, submitted by readers, published in Truth (London, England) of Wednesday 13^th May 1914:

Not half. My word, if I catch you bending. Get your hair cut. Chase me? Have a drop of gin, old dear? Tell your mother that’ll be ninepence. Wait and see. I should say so. Have a banana. Thank your mother for the rabbit.—Bridget.

4-: From Vets., an article by J. Cahara about army veterinary surgeons, published in The Illustrated Sporting and Dramatic News (London, England) of Saturday 1^st January 1916—in the following passage, the author tells what happened after a shoeing-smith played a practical joke on an Irish vet:

It came as a sudden surprise to the said shoeing-smith, and caused that worthy utter embarrassment and the rest of the squadron great joy when, on parting, the vet. turned to him and said:
“Well, Johnson, that seems all right. Good-bye. Thank your mother for the rabbit.”
His wits—strange to relate!—saw no meaning in this, and accordingly thought they had been scored off. He is still wondering what connection this has with anything, and is very, very polite to the vet.

5-: From Army Sayings Illustrated, by Private. R. B. Ogle, published in The Gazette of the 3^rd London General Hospital Wandsworth (London, England) of February 1918:

“. . . . And thank your Mother for the Rabbit.”

6-: From Ulysses (1922), by the Irish author James Joyce (1882-1941)—Zoe Higgins is a Yorkshire girl:

ZOE: The devil is in that door.
(A male form passes down the creaking staircase and is heard taking the waterproof and hat from the rack. Bloom starts forward involuntarily and, half closing the door as he passes, takes the chocolate from his pocket and offers it nervously to Zoe.)
ZOE: (Sniffs his hair briskly) Hmmm! Thank your mother for the rabbits. I’m very fond of what I like.

7-: From London Farewells, published in The Sporting Times (London, England) of Saturday 6^th November 1926:

Among London’s toilers partings are frequent; and there is always some jolly catch phrase to take the edge off by raising a smile.
[…]
“Remember me to Jessie and Minnie. And mind you give my love to Cyril.”
Interesting to nobody but persons concerned; but ears are pricked when the Cockney worker fills the station with a robust:
“Good-bye; an’ thank your mother for the rabbit.”
Mysterious, impalpable rabbit—was it alive or dead, meant for [two illegible letters]able or table? Non-existent bunny, you have softened many a poignant parting!