Male children of Gull-Skeggi have the surname Skeggvason (their father's full given name is Skeggvar). Female daughters of Gudrun, the matron of the house, have the surname Gudrunsdottir. Thralls have no surname. People who don't fit into one of these categories either have a surname from some other family in the area, or a Deed Name based on something they did.
Skeggvar is called Gull-Skeggi because he was as peripatetic as a seagull during his years of viking. Or so he says now.
This is generally how family names work in the Lands of the Linnorm Kings: sons have last names based on their father's first name, and daughters have last names based on their mother's first name. Once someone gains their own fame, they get a use-name (like "Gull-Skeggi") that replaces their full name (in his case, "Skeggvar Arneson").