Fanny is usually a pet form of
Frances. It was rather popular in the time period you mentioned, both as
Fanny and
Fannie. When "fanny" somehow became an euphemism for "buttocks", the name dropped out of usage. A famous bearer is cookbook author
Fannie Farmer (1857-1915) (here's an online version of her cookbook:
http://snipurl.com/99rw).
Click on the hyperlinks for more info on the name.
Miranda