"There are many men in London, you know, who, some from shyness, some from misanthropy, have no wish for the company of their fellows. Yet they are not averse to comfortable chairs and the latest periodicals. It is for the convenience of these that the Diogenes Club was started, and it now contains the most unsociable and unclubable men in town. No member is permitted to take the least notice of any other one. Save in the Stranger's Room, no talking is, under any circumstances, allowed, and three offences, if brought to the notice of the committee, render the talker liable to expulsion. My brother was one of the founders, and I have myself found it a very soothing atmosphere."
We had reached Pall Mall as we talked, and were walking down it from the St. James's end. Sherlock Holmes stopped at a door some little distance from the Carlton, and, cautioning me not to speak, he led the way into the hall. Through the glass paneling I caught a glimpse of a large and luxurious room, in which a considerable number of men were sitting about and reading papers, each in his own little nook.
-Arthur Conan Doyle, The Greek Interpreter
The Diogenes Club is an institution which exists in the fictional Victorian London of Sherlock Holmes. As the above passage tells us, it was co-founded by Sherlock's brother Mycroft Holmes, and with a very few notable exceptions, it is the only place he is seen in the stories - when he is seen outside of his lodgings, his office or the Club, it is a sign of great disturbance in the natural order of things.
Although Conan Doyle (via Sherlock) does not touch on the origin of the Club's name, some guesses can easily be made. Diogenes of Sinope was one of the founders of the philosophy of Cynicism, but that is probably not the whole story. Rather, later on in his life Diogenes was captured and sold into slavery, where he was purchased by a Corinthian named Xeniades as a tutor for his sons. While in Corinth, Diogenes came to believe strongly that artificial society was the cause of much unhappiness, and his ideas on the rejection of materialism and the identification of a man with a polity (he is credited with the origin of cosmopolitan by naming himself a cosmopolites or citizen of the world and thus no single polity) were such that the Stoics would later cite him as a man of wisdom. Sources state that his later life saw him preach the 'virtues of self-control' which is probably the most direct link to our purpose.
Given his rejection of society and his vision of himself as external to it and its intercourse, Diogenes begins to make perfect sense as the namesake of Mycroft's club.
Sources:
Philosophy Basics on Diogenes
Wikipedia on Diogenes
Gutenberg's text of The Greek Interpreter