Half right. The Icelandic phone books are organized by "given name" but not instead of "surname". They don't have surnames. The "son of" or "daughter of" is based on the father's given name. So it doesn't get passed down.

If Hrolf has a son Eric he would be called Eric Hrolfson. But if Eric has a son Leif, he would be called Leif Erikson.

At least people in that culture don't change names at life events! (there are cultures where this is common). We only slightly do that and whether we recognize that we need to consider "real name" how we are addresed as opposed to what might be on a piece of paper. Thus our babies are often given "official" given names that will serve as their given names only in adulthood if at all but in the intervening years are called something else.

I know all that, but probably my English (my third language) is failing me. If Hrolfson is not "surname", what is it?

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It's a patronymic. Surnames get handed down, whereas a patronymic doesn't get handed down the family lineage in the same way. Sometimes, historically, patronymics have turned into surnames, and some cultures use both (Russia, for example).

More info: http://en.wikipedia.org/wiki/Patronymic

Or the simple general English term for that, and surnames, is last name, though that label doesn't work so well for Asian names where the family name comes first.