How to find the non-anglicanized italian name of john gillam (1550+-25yrs.)?

i recently found out from my dad, that we have italian ancestry. i was told that a friend of his helped look us up, and found we had italian family (sicilian) ancestors. i've gone back as far as the mid-1500's, and i found a john gillam. and, according to british records, he arrived in the 1500's, was italian ethnic, and italian of origin. when a person comes from a non-english speaking country to a english one, the name usually gets Anglicized. john deviously would be giovanni, but the last name of gillam has been Anglicized. i'm trying to make the full name, because i know there are italian gillam's out there, and i want to connect with my italian roots.