Reykjavík City Hostel

The hostel's back porch. You can rent bikes.
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I stayed at the Hostelling International Reykjavik city hostel and it was really great. I'd recommend it. Very clean, friendly staff, not too far from downtown (~25 minute walk), free wireless Internet, a nice lounge/common area, all lodging services you can think of are here (kitchen, fridge, cooking range, laundry, ping-pong, storage, foosball, DVD player, books, cable television, etc.).

If you're a Hostelling International member, you can get a room for ISK 1700 ($28) per night, and an extra ISK 400 ($7) if you're not a member. If you have camping gear, you can camp for ISK 800 ($13) per person, per night.

I met people here from all over the world: a group of guys originally from Australia that had just finished a year of studying in Sweden; a girl that grew up in Paris, went to part of high school and college in the states and just finished a year studying in Italy; some hardcore rockers from Denmark; more hardcore rockers from Boston, MA.