Cheaper accommodation in London tends to be concentrated within certain areas, such as Victoria and Kings Cross (both of which are only a few minutes away from Covent Garden on the Tube). Covent Garden itself has hardly any hotels or B&Bs.
However the Strand Continental has received lots of positive comments here on AB, and it's right on the edge of Covent Garden:
http://www.strand-continental.co.uk/Welcome.html
(Several people have stated that they've booked at lower than the hotel's advertised 'rack rates' by looking for the hotel on 'late booking' websites).