Wordorigins.org

This is an American slang word of that dates to the late 19th century. It probably comes from fawney, a slang term of earlier vintage, which in turn is from the Irish fáine, or ring. Fawney also lent its name to a confidence game, referred to as the fawney rig or going on the fawney. From George Parker’s 1781 A View of Society and Manners in High and Low Life:

There is a large shop in London where these kind of rings are sold, for the purpose of going on the Fawney.