It seems to have been named after Diogenes the Cynic (though this is never explained in the original stories) and was co-founded by Sherlock's indolent elder brother Mycroft Holmes.
The club as described by Sherlock Holmes in "The Adventure of the Greek Interpreter": "There are many men in London, you know, who, some from shyness, some from misanthropy, have no wish for the company of their fellows.
It is for the convenience of these that the Diogenes Club was started, and it now contains the most unsociable and unclubbable men in town.
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.
This may have its root in "The Adventure of the Bruce-Partington Plans" (1908), in which Mycroft Holmes is revealed to be the supreme and indispensable brain-trust behind the British government, who pieces together collective government secrets and offers advice on the best way to act.