My Italian name dictionary (
Santi e Fanti by
Enzo La
Stella T.) says that Dìlio is a short form of Edìlio. The origin of Edìlio is uncertain, but La
Stella says it's possibly from Latin
aedilis (a title of an elected official in ancient Rome), or possibly just a diminutive of Germanic names starting with
Ed- such as
Edwin. Since in Italian it is normal to turn a male name ending in -o to a female name by changing the -o to -a, I would assume that Dìlia is the feminine form of Dìlio.