Icelandic has -dóttir and -son endings and they do mark whose child you are, true. One note though - Icelandic has more genitive endings than just -s, so you get all sorts of patro- and matronymics, our Árni being a good example: his son is not Árnisson but Árn
ason. Likewise if a woman called Laufey had a baby and decided to give it a matronymic last name it would be called Laufey
jarson/Laufey
jardóttir, all depends on the declination of the particular name.
Icelanders are not allowed to have actual surnames as per the naming committee's rules, unless you're born into one of those few families that took a surname during the short time window when that was possible. That's why Jón Gnarr had to leave country to officially change his name to Jón Gnarr. These families are really rare however and I think their names must all be listed somewhere... plus since Icelanders don't take on the spouse's last name when they marry (for obvious reasons) some of these names may even have died out long ago.
I can help out with Icelandic names if it's needed by the way, just PM me.