Stay at the edge of the city, the hotels are cheaper there. I would recommend Uxbridge. If London was a clock face, Uxbridge would be where the number 10 is. There is a hotel next to the train station and the station has two lines that branch off from it. The chain hotels don't check or care how many people are in a room so you can bring inflatable mattresses and bring a whole family to one double room.
If London was a clock face, Heathrow airport would be a polca dot beyond the clock face next to the number 9 so there are several buses that go from Heathrow to Uxbridge, the Carousel 102 to high Wycombe passes through Uxbridge.
If you did stay at Uxbridge, there are several perks that come with the train station, both the Piccadilly and Metropolitan lines stop at that station.
If you stay on the metropolitan line and get off at Finchley road, you can get a Jubilee train from the same platform because jubilee and metropolitan line trains share a platform and go in the same direction. The Jubilee line stops at some key points on the river, by the clock tower with the big Ben bell in it and near London eye. The Jubilee line also stops at the Greenwich ("gren itch") peninsula. Most the interesting stuff is near the river, even Trafalgar Square is a 15 minute walk from the river Thames ("tems"). The Piccadilly line goes to most the important places, Leicester Square, Piccadilly Circus and places like that.