You used to be required to adopt an Icelandic forename - not surname. You still kept your original name (if you wanted) and it was up to you which one you used on practice.

But as sibling comment says, that’s been dropped.

For surnames, anyone (including Icelanders) is allowed to use a family name if they have a claim to one, which is defined as having a parent or a grandparent carrying that name as a surname. So foreigners with family names can pass those to their children if they like and skip the patronym.