The name Fanny (and similar names like Fanya and Fani) is a newcomer. We have a relatively complete set of birth records for Nezhin Ukraine 1860-1918. The first Fanny is born in 1891; there are 6 total in the 1890s (out of roughly 2500 births), more in the decade 1900-1909; and even more in the period 1910-1918, even though the total births in the latter period has shrunk to about 1400.
By contrast, girls are named Feiga throughout the 59 year period.
