English of unclear origin.

ONC gives the derivation as of Greek origin, referring to the Island of Delos.
Whyte suggests that Delia originated as a shortened form of Bedelia which is a view shared by many correspondents in the US.  Delia is found as a pet name for Bridget which is suggested as a variant. of Bedelia.

Neither of these main sources considers Delia likely to be an abbreviation of Cordelia which appears to have probably been a name invented by Shakespeare in King Lear.

Delia is not found in the 1841 Census of Aberdeenshire.