Do you remember in The Player when studio honcho Griffin (Tim Robbins) takes the Icelandic girlfriend (Greta Scacchi) of the screenwriter he killed to a museum party? The Hollywood people don't know her, and someone goes "Whose daughter did he say she was?" Funny because true.
A friend from Iceland and his wife just had a baby (congratulations, Snorri & Lydia!), and he's posted an short explanation of the Icelandic naming system. It's patronymic (and sometimes matronymic), which means that a kid's surname isn't a family name, but a "son of whoever" and "daughter of whoever" indicator.
All of Scandinavia used to use a similar system, but they stopped at some point, which is why Minnesota is full of so many Gustafsons, and why Utah is freaking with Christensens. Iceland, meanwhile, is still mixing it up.
Icelandic last names [sturluson.com/snorrason]
Icelandic name system [wikipedia]
[update: while looking around for the official book of names for Iceland, I found NordicNames.de.]

When we were in Iceland we were talking to this Pakistani guy who became a naturalized citizen, nad he said that up until a few years ago, they used to make you take a traditional Icelandic name when you became a citizen. So this guy would have become Thor Gudmundson or whatever, which he thought was hilarious. Oh, and Iceland phonebooks are apparently sorted by first name instead of last name.
Lived in Iceland for 3 years. Great country. I would love to visit again.
All children born in Iceland must be given a traditional Icelandic name as published in the book of names, called the Nafnabokin (name book). If the name you want to give your child isn't in The Book you have to petition the name commitee, which meets quarterly to review and approve name petitions.
Amanda, that book sounds pretty good... I think we need something like that here in the U.S., only it shall say that people must give their children a name that a human can live with and it MUST be spelled correctly. No more adding y's and i's where they don't belong! And NO fruits or beverages allowed (though I'm still considering Jack Daniel for a boy *wink*)... heck the list could go on and on!
I was born Uma Halgrimmsdottir, which means daughter of the half ugly!!
Thank goodness my mother married again, and again actually.......yeah mom!