Irish, Scottish and English, Anglicized form of Brighid, the Celtic fire goddess or from Old French Brigitte.
Brigitta was used in old documents in Latin for these names. Search on Brigitt* to allow for Latin case endings.
A correspondent [LG] has evidence of Elizabeth or the pet name Betsy being used for Bridget in a family of Irish extraction:
"My ancestor in the early 19th century was recorded as Bridget on some statutory records and as Elizabeth on others. For some years I actually thought I was looking for two separate individuals. Having an unusual surname for this area helped. I later discovered most of the Bridgets in my family (1819-1930), although registered as Bridget, were known as Elizabeth or Betsy. These names might be a family quirk but perhaps it could happen in other families of Irish extraction."
This usage has been supported by another correspondent [KC2] with Irish ancestors.
Several correspondents, particularly in the US and Canada, have noted that Delia is a common pet name for Bridget, being a shortened form of its probable variant Bedelia.
A Canadian corespondent [JL5] has examples of one Bridget in Ireland who was known only as Delia in the US and another who was called Elizabeth or Eliza in the US until she died when her birth name was registered.