Depends on which terminal. Heathrow Express is less useful for T4 and there's a decent amount of walking at Paddington. I'd probably either take Victoria to Green Park and Picadilly from there, or walk to King's Cross and Picadilly all the way.
Travel lodge at Covent Garden. Then Picadilly line to Heathrow.
Thanks for the suggestion but I'm looking for something much cheaper. easyHotel is £35/night.
Any other suggestions?
There are some fairly cheap places around Earls Court. That would place you on the Picadilly line for Heathrow.