If your main objective in finding London digs is Piccadilly Circus proximity,
Piccadilly Backpacker's Hostel, where dorm bunks can be had for around $20 / night, fits the bill on teeny Sherwood Street, steps from the west exit of the
Piccadilly Circus tube stop. Single room more your style? Head to the Regent Palace Hotel instead, practically across the street and light years up the budget hotel ladder. And if cheap sleep in a decent hostel is your chief goal, look into other London hostels where the experienced staff is a little friendlier and floors are vacuumed periodically, like the
Wake Up London hostel in Paddington, minutes away by tube or bus.
Check-in at the Piccadilly Backpacker's Hostel is a drag -- you must be okayed by sentries inside the street-level front door and then lug your luggage to the second floor, where you'll get in line to repeat yourself. If you're checking in before 1:00 p.m., you can leave your stuff in the non-secure luggage storage which, as a less than polite employee told me, is free.