The verb Bunbury means: to use of an imaginary person as a fictitious excuse for visiting a place or avoiding obligations.

Hence, the verbal noun Bunburying denotes the use of an imaginary person as a fictitious excuse for visiting a place or avoiding obligations.

This verb and verbal noun refer to Bunbury, the name of an imaginary character in The Importance of being Earnest: a trivial comedy for serious people (London: Leonard Smithers and Co, 1899), by the Irish author and wit Oscar Wilde (1854-1900), first performed on Thursday 14^th February 1895 at the St James’s Theatre in London.

The following dialogue between Algernon Moncrieff and Jack Worthing takes place at the beginning of The Importance of being Earnest:

ALGERNON I have always suspected you of being a confirmed and secret Bunburyist; and I am quite sure of it now.
JACK Bunburyist? What on earth do you mean by a Bunburyist?
ALGERNON I’ll reveal to you the meaning of that incomparable expression as soon as you are kind enough to inform me why you are Ernest in town and Jack in the country.
[…]
ALGERNON You have invented a very useful younger brother called Ernest, in order that you may be able to come up to town as often as you like. I have invented an invaluable permanent invalid called Bunbury, in order that I may be able to go down into the country whenever I choose. Bunbury is perfectly invaluable.

The verb Bunbury and the verbal noun Bunburying occur several times in The Importance of being Earnest. The following extracts, for example, are from Act II:

JACK This Bunburying, as you call it, has not been a great success for you.
[…]
JACK Well, you’ve no right whatsoever to Bunbury here.
ALGERNON That is absurd. One has a right to Bunbury anywhere one chooses.

In the review of The Importance of being Earnest, in the column Stage Whispers, published in The Umpire (Manchester, Lancashire, England) of Sunday 17^th February 1895, three days only after the première, ‘The Prompter’ noted that Oscar Wilde had enriched the language with the word Bunburying:

Some of the epigrams in this piece are as evanescent as soap bubbles, but they are thrown off in such profusion that they make a brilliant display, and after curtain-fall remain a pleasing memory.
But I am neglecting the story, of which, to tell the truth, there is very little, and that little preposterous.
Its best effect may be to enrich the language with a word—“Bunburying.” You don’t know what “Bunburying” is? No more did I—the word I mean; the thing has long been known.
“Bunburying” is to have a mysterious relation of the name of Bunbury, who always falls ill and requires to be visited. Either when you want to go out of town on some unavoidable business, or when you receive an invitation to dinner which you wish to decline, there is always that invitation to Bunbury which comes in handy.

I have found an early use of the verbal noun Bunburying in The Diary of a Country Gentleman, by ‘W. J. M.’, published in The County Gentleman, Sporting Gazette, Agricultural Journal, and “The Man about Town” (London, England) of Saturday 30^th March 1895—i.e., a month and a half after the première of The Importance of being Earnest:

“BUNBURYING.”

Thursday.—Though my Bunbury is a very real one, in whose interests I am paying my quarterly visit to his school to inquire into the working and prospects of the Army Class. Having interviewed all his pastors and masters, and having ascertained that they passed five out of twelve into Sandhurst at the last exam., I am taken off to see the school boxing and gymnastics.

‘G. N.’ used the phrase to go Bunburying (meaning, apparently, to go and see The Importance of being Earnest) in the review of Noël Coward’s Hay Fever ¹, produced by the Leicester Drama Society—review published in the Leicester Mercury (Leicester, Leicestershire, England) of Tuesday 6^th April 1937:

This play [i.e., Noël Coward’s Hay Fever] may possibly claim to be the best English light comedy since the ’nineties went bunburying. Indeed (but let this be whispered very gently, lest the legion of the bitter sweet ride down upon me in awful cavalcade) another age may rise which will claim this to be by far the most considerable contribution made to the theatre (as yet) by that brilliant ward of Thespis, Mr. Coward.

¹ Hay Fever (1924) is a comic play by the British playwright, actor and composer Noël Coward (1899-1973).

The verbal noun Bunburying occurs in the following theatrical review, published in The Illustrated London News (London, England) of Saturday 13^th April 1940:

“GOOD MEN SLEEP AT HOME,” ²
AT THE SHAFTESBURY.

Wherever good men sleep, it won’t be at the Shaftesbury. A jolly farce, calculated to keep even over-dined bad men awake. The author, Mr. Walter Ellis, owes a lot to his producer, Mr. Walter Ellis. There is also a lot owing to “The Importance of Being Earnest,” for “Bunburying” is a feature of the play. Mr. Hugh Wakefield, Mr. Mackenzie Ward, and Miss Olga Lindo make an ideal team. One hopes to see them again in another farce. But it will not be soon. This one should run for months.

² The following summary of Good Men Sleep at Home, by the British playwright Walter Ellis (1874-1956), is from Concord Theatricals:

Walter Ellis undoubtedly has the knack of producing farce that is more than comedy. His chief character is a financier, George Warburton, who takes the precaution of keeping his wealth in his wife’s name, but lacks the same sense of security in his social habits.
When George is involved in a motor accident while driving with the glamorous film star, Clare Boy, and finds that he and a woman novelist, mistaken for husband and wife, and both unconscious, have spent the night in a country roadhouse, the entanglements begin.
George’s wife is soon on the trail, and one of his clients also sees an opportunity to “square up” some financial deals. The woman novelist and the film star get poor George into a black position before he finally wins the return of his wife’s affection with a feigned attack of insanity.

The verbal noun Bunburying occurs in the review of two biographies of the British landscape painter Joseph Mallord William Turner (1775-1851), Standing in the Sun: A Life of J. M. W. Turner (London: Sinclair-Stevenson, 1997), by Anthony Bailey, and Turner: A Life (London: Hodder & Stoughton, 1997), by James Hamilton—review by Peter Conrad, published in The Independent (London, England) of Sunday 29^th June 1997:

Bailey and Hamilton also diverge on Turner’s elderly liaison with the Margate landlady Mrs Booth. He secretly cohabited with her, adopting the name of her late husband as an incognito. Neighbours took him for a retired admiral, and fondly nicknamed him Puggy Booth. She looked after him in his final illness, spending her own money—despite his wealth—in the process. Bailey treats the episode as an exercise in Bunburying, disclosing Turner’s talent for subterfuge while making clear his sexual appetite, and has no doubts that Turner warmed the widow’s bed. Hamilton is more delicate. He insists that Mrs Booth played a “motherly role”, since Turner’s only “darlings” (by his own admission) were his paintings, and he absolves the couple of “any intention to deceive … or to run a double life”.

An interesting use of we’re going off Bunburying is mentioned in the column Peterborough, published in The Daily Telegraph (London, England) of Friday 4^th May 1990:

Wilde life
A CENTURY after London society first applauded Oscar Wilde’s polished aphorisms, snatches of the playwright’s work can be heard in the most unlikely setting: a remote village in the Sudan.
Tim Allen, a research fellow of St Anne’s College, Oxford, has just written a paper about his experiences teaching in the village of Atar, where old favourites on the national English syllabus have provoked mixed reactions.
The tribe’s taboo against incest includes the marriage of cousins, so Wuthering Heights was regarded as scandalous in the extreme. But Wilde’s The Importance of Being Earnest was a hit.
Local youths were so intrigued by the dialogue that bits of it have passed into their conversation. When they plan a journey to the next encampment in search of girls, for example, they nudge each other and whisper: “We’re going off Bunburying!”