"Bunbury" is the imaginary friend Jack (Earnest) must "visit" in order to avoid attending his aunt's long and boring dinner parties. This is a purely virtual invention of his to have an excuse for his absence. According to Jack, his "friend" is of very fragile health and often needs his personal attendance. Of course, these "spells" conveniently occur whenever Jack needs them to get away.

On a symbolic level Bunbury represents all the sham and double talk that has got Jack into trouble in the first place. It is the main vehicle for the intrique of the story- along with, of course, his own lost (then refound) identity.

Bunburying has several connotations:

It was said by Aliester Crowley, according to Neil McKenna, that Oscar Wilde once "took the coach to Banbury, met a Public School boy there, and the agreed to meet in Sunbury" for a secret meeting.  Hence, the word "Bunbury" was coined by Oscar and his set as a term that means "leading a double life" or "hiding certain secrets".

Bunbury was then the name Oscar chose to name the non-existent character of the invalid that Algy goes to visit each time he also wants to go somewhere else and do whatever it is that he does.

Hence, the connotation is double, as most of the symbols, names, and epigrams said in Earnest are also double entendres for Oscar's own tribulations.

Earnestness, also according to McKenna might be a Victorian code name that is a bad translation from the French of the term "Uraniste" or Uranian (homosexual) Earnest/Uraniste.  Taking it from that view, then the entire play takes a different perspective.

Yet, if we use the term "as is", bunburying would be the antithesis of earnestness, that is, lies vs. truth. In those days, however, the truth "was never pure and rarely simple" and the hypocritical, uber prudish and elitist Victorian mentality permeated just about every aspect of morality and righteousness. 

Everyone bunburied amidst the societal insistence of earnestness.

Algy invents a friend "Bunbury" who is an invalid, in that way allowing Algy to get out of boring dinners with family friends and relatives. When he wants to get out of something, he tells his family that he has to go care for the invalid. He calls this "bunburying" obviously for the name of his made up invalid, bunbury. What this says about Victorian earnestness is that friendship is important, and everyone believes that tending to an invalid is a good excuse for leaving at the last minute. It's a critique at Victorian society. Everyone is so earnest about caring for relatives and invalids, they just assume that Algy is the same, and accept it at face value.