"The Blue-Stocking Club" was a literary
salon founded towards the end of the 18th century by Elizabeth Montagu, a
society hostess and
intellectual. The gatherings were exclusively of
literary and educated women, with only a select number of males invited to attend (among them
Joshua Reynolds and and
William Wilberforce).
The names itself derives from a poem written by one of the members of the club, Hannah More, describing their meetings and entitled Bas-Bleu (blue stocking). I do not know if the explanation of the name given by Webster here is correct, apocryphal or somehow related to the poem, but however the existence of the poem is fact.
The term "bluestocking" was not always derogatory. The Regency and the period preceding it, the end of the 18th century, both placed a very high premium on wit and elocution as well as literacy. Women during the Regency period were much better educated in general than their Victorian counterparts, and it is only as the 19th century, with its backlash against "decadent" Regency values, wore on that brains became a marriage liability in a female.