Where should I stay?

I get cheap "for London" hotel accommodations using Hotwire next to (well very near) Heathrow. Last time it was the Sheraton for o/a $85. The next day you can tube into London after rush hour pretty cheaply, have a nice day, and then return to your hotel in the evening(figure an hour each way). This has the added advantage of positioning you near the airport for your next day departure. If you're leaving out of Terminal 5 on BA it goes pretty smoothly. Otherwise Heathrow remains a traveler's nightmare.

If you will arrive at LHR in the morning, I recommend that you immediately go into London and stay there both nights. Most hotels will keep your bags for you until you return to check in. You can do a little sightseeing in the afternoon, staying on your feet to fight jet lag, and then go to bed at a reasonable hour (London time). You'll wake up with your body clock adjusted and ready for a whole day to explore London. The Tube starts running early enough in the morning that you can easily get to LHR in time to meet your check-in deadline. Travel time between LHR and central London is around an hour. Buy an Oyster card at a LHR Tube station, load it with about £20, and use it as a pay-as-you-go card. When you get back to LHR you can get a refund on both the card deposit and any money left unspent.

The Arran House Hotel is a B&B in Bloomsbury whose regular rooms aren't cheap, but it has some dorm-type rooms at hostel-like prices. All guests get a full English breakfast.

On the same street as Arran House and just a few doors away, is the Ridgemount Hotel, much less expensive and well located within walking distance of the Br.Museum, the John Soanes Museum, etc. It is located on Gower Street.