Sicilian surnames starting with Im- or In- generally refer to a founding widow.
In this case Mr Bruglia died and his wife became head of the family. In Sicilian usage she would have been known by her maiden name plus “In Bruglia”, later surnamed Imbruglia for the children.
Several such surnames exist, e.g. Ingrassia, Incalcaterra, Inpinna, Ingrasciotta, Indelicato, each pointing to a widowing event.
There are exception, e.g. Indemburgo is from the Prussian Hindemburg, not from a widowed Signura In Demburgo 🫣