1. Irish Gaelic, "fair", Anglicized as Fion/Fionn.
A correspondent [WF] reports that those with the Irish Gaelic names Finin, Finn, Finian, Fineen & Fyneen were forced to use the Anglicized form Florence [q.v.] by British Rule.
2. Scandinavian, from Old Norse, either for someone from Finland, or from a short form of names containing the syllable -fin- "Finnish".