I was in Iceland a few years ago so my info on hotels is not the most updated.
I stayed at the Radisson Saga. Not the most central of the Radisson hotels in Reykjavik but cheap and really good.
Reykjavik is pretty small and in a day you can see all there is to see. A soak in the Blue Lagoon is absolutely not to be missed and the place is a very short drive from the city, something like 20 minutes if I remember right.
If you stay more than one day I suggest renting a car or taking a bus day trip and do what they call the Golden Circle (Thingvellir, Gullfoss and Geysir) as that will give a taste of what Iceland can be.