If you want dead centre, it's hard to get cheap.
About the cheapest would be the Travelodge (York Central), it's at the end of Piccadilly and it's an easy walk into the centre from there. There is also a Quality Inn (Hotel 53) on Piccadilly as well but dont know about pricing. Watch out though as hotel parking is scarce in the centre.
You could also try the Ramada Jarvis on Tadcaster Road, not very nice but often does cheap deals and very well placed for the pubs and bars on Mickelgate.