You can only name your child from a list of approved names, there's restrictions on certain names that can either only be used as first or second names (sometimes those internet name searching things fail because they can't tell the difference), the spelling has to go by Icelandic grammar, you have to usually pick a name that's in the same gender as the child grammatically, and also there are absolutely NO gender neutral names - they go against the law. You can't use foreign spelling, meaning that even in the off-chance that the naming committee allows the child a foreign name it still has to be spelled in an Icelandic way (and cannot have letters that are not in the Icelandic alphabet, such as C).
To be allowed a foreign name the child has to have at least one parent of foreign ancestry, and over here that means that the parent must have been born outside of Iceland and have lived elsewhere for most of their life. The foreign name applications are gone over case by case and it's not at all guaranteed that they'll be approved, plus you'll have to apply specifically for the name to be considered and pay some handling fees. Name not approved? Back to the drawing table for you, you'll just have to apply with a new one and pay the fees again... or risk your child be forever called "girl" or "boy" in official documents.
Lately the former mayor of Reykjavík, Jón Gnarr, wanted to change his name to be Jón Gnarr for real since it's not his birth name, however, it's the only name he's been using for years. Naming committee gave him a strict no so he went to the USA to change it... and upon returning got the same no again. There was also the matter of Blær (= Breeze) who wasn't allowed her own name for most of her life despite the fact that it is a real Icelandic girl's name, you can f.ex. find a character called Blær in a book by Laxness... but it had been a long time since the last Blær was born and the name declines in masculine. She's a good example of what it means that the name has to match the child's gender grammatically, not only in reality (wow that felt a bit silly to write).
I'm not against some naming laws, mind you, it's good that not just anything is allowed (such as that one couple that tried to name their daughter Lúsiferia) but I find that the Icelandic naming laws are unnecessarily strict.
