Generally speaking a patronym is the social custom, but a matronym can be decided upon by the parents, probably most common in the case of single mothers though. Then if you so fancy you as an adult can choose to bear either a matronym or a patronym. Some even go for both.

But bear in mind that the patro- or matronym is not really a part of one's name. It is more of a social identifier, but when foreigners address Icelanders by last name, they come off really idiotic. Your given name is who you are, you as a person. The rest is just bits that relate you to those that contributed to your making. I personally dread the day when someone is going to refer to me by my last name, and that is going to happen soon, me being a scientist and all. It will be odd to have this bizarre custom forced upon me.

Historically the patro- or matronym was not even a part of one's name, people just had their given name, and then when further explanation was needed of who they are a long list of their lineage could be rattled off in the form of "Hrafnkell Hjálmarsson Auðarson Grímsdóttur". Indicating that Hrafnkell's father is Hjálmar, who was the son of Auður, who was the daughter of Grímur. Then Christianity came and with it gradually came the culture of continental Europe, and a patronym became the social norm.