In reply to PMG:
There are a range of car rentals to go for and there's probably no right choice.
Definitely check the range of insurances you can get - especially if you plan on going into the highlands.
I would recommend either SagaCars, citycarrental or sadcars just because I've had experience with each.
As mentioned before, question yourself whether you need a rental car. Lots of places are accessible easily by hitch hiking or buses. If you wanted to get into the highlands, there's a bus to Thorsmork for example. Or Kerlingafjoll I would recommend!
Just spent a year there, so if you want any more recommendations, feel free to drop me a msg.