I highly recommend International Students House, a combination hostel/student residence right next to the Great Portland Street tube, near Regent's Park and a short walk down Great Portland to Oxford Circus and SoHo. I spent a lot of time there when I lived in London and have stayed for as long as 2 weeks. They have a great bar, a lively social scene and people in and out from all over the world. It is also the place you are the least likely to get your stuff stolen, because the professionalism of the staff seems to spill over.
You can get a dorm for about 12 pounds a night, but the better deal is the quad room where there will be no more than 4 and sometimes just you for 18.