Depends on what you mean by cheap.
I have been looking as well - there are a few 'short time' hotels in the Malate area off Quirino Avenue - the Soho Hotel and Victoria hotel - for 1,200 pesos a night. These are basically 2 star hotels - single room, bathroom, cable tv, air con, double bed. Nothing flash
But all the three star hotels are around the 1800-2000 pesos mark. There might be a few around the 1,600 pesos mark. I know that the Gran Prix hotel in Pasay - near the bus terminal and 30 minutes drive from the airport cost 1,400 pesos for a normal room.
I was thinking to get an apartelle - which would cost 1,600 pesos a day. But its hard to find anything close to the airport.
Apart from that everything else is around the 4,000 pesos cost per night. Just depends on what your budget is.