I lived in Iceland as a Navy wife for a couple of years.If I recall, the last names of the Icelanders is their father’s name plus dottir (if they are a girl) and son of they are a boy. So if your Dad’s first name is John and you were a girl, your last name would be Blaer Johnsdottir. If you were a boy it would be whatever Johnson.
I think it would be more important to keep the boy’s names traditional to preserve the culture.
Johnsson