English, from the surname, originally for someone from the city of Derby or from West Derby near Liverpool. Derby [pronounced as "Darby" in Britain] is from Old Norse for "deer settlement".
The Irish Gaelic name Diarmait [and variants, see above] may be Anglicized as Darby although Dermot is more usual. Darby may be found as a pet name for Dermot. These names were all ultimately derived from Demetrius [q.v.]. In an Irish family a correspondent [GGF] has reported Darby and Jeremiah being used interchangeably.