Iceland, in common with the other Scandinavian countries, has a patronymic system of surnaming. Helgi, the son of Jon Asmundson, is known as Helgi Jonsson, not Helgi Asmundson. His sister Helga has a different surname - Jonsdottir. Basically, these surnames mean 'Helgi, son of Jon' and 'Helga, daughter of Jon'. Jon's father was Asmund. This system may seem confusing, but most Icelanders (with a little help from the extensive written records) can trace their ancestry back to the settlement of Iceland.
Rarely, a family may take a surname from a place. Laxdals, who settled in the Elfros area, are an example of this.
If you want to create your own Icelandic surname, add 'sson' or 'sdottir' to your father's first name. Examples: