The phrase "Pretty Fanny's Way" has gone out of fashion. It is attributed to the Graveyard poet Thomas Parnell, and might be replaced by the word eccentric today.
Parnell, in fact, is considered the first of the Graveyard poets, with his "A Night-Piece on Death" (1721) considered the first representative of this school of poetry. Today's post line is part of a verse that runs 'And all that’s madly wild, or oddly gay,/We call it only pretty Fanny’s way’.
Walter Scott used the phrase in his journal entry of November 28, 1825. I found the post interesting for it's practical consideration of how best to socialize. Per Scott, '[John Gibson] Lockhart must be liked where his good qualities are known, and where his fund of information has room to be displayed. But, notwithstanding a handsome exterior and face, I am not sure he will succeed in London Society; he sometimes reverses the proverb, and gives the volte strette e pensiere sciolti, withdraws his attention from the company, or attaches himself to some individual, gets into a corner, and seems to be quizzing the rest. This is the want of early habits of being in society, and a life led much at college. Nothing is, however, so popular, and so deservedly so, as to take an interest in whatever is going forward in society. A wise man always finds his account in it, and will receive information and fresh views of life even in the society of fools. Abstain from society altogether when you are not able to play some part in it. This reserve, and a sort of Hidalgo air joined to his character as a satirist, have done the best-humoured fellow in the world some injury in the opinion of Edinburgh folks. In London it is of less consequence whether he please in general society or not, since if he can establish himself as a genius it will only be called "Pretty Fanny's Way."