The City Hostel mentioned earlier had no private rooms available during my dates (6/18-6/22).
http://www.hostel.is/heimilin/Reykjavik/Eindex.htm
Good reviews from backpackers at:
http://www.bugeurope.com/reviews/Ice...ykjavik/is023/

So I booked the Hotel Gardur, a university dorm that converts to a guesthouse in the summer. $110 for a double room, but no private bathroom.
http://www.icelandichotels.is/xodus.asp?id=191
Hotel Gardur is pricey, but it looks nicer than the other, more expensive, options for private rooms at Hostels.com:
http://www.hostels.com/en/findabed.p...ountry=Iceland