Now that you've landed in London and got your preferred mode of transportation sorted, you'll need a place to lay your head at night.
The cheapest options would be hostels. I have no experience with those so I really can't do much more than point you to London Hostel Association. There's lots of other hostel booking sites as well, Google is your friend.
Another inexpensive option for summer is the student accommodations at the various universities around London. University College London (UCL) has a variety of options as does many of the colleges affiliated with the University of London. Again, Google is your friend. There isn't one central spot to check which of their colleges have summer accommodations.
Budget hotel chains might be another good option. They are nothing particularly special, usually pretty generic but they're clean and dependable. They may not be right in the centre of town but they are usually close to an underground or bus route or two. These include Ibis and Travelodge. Sometimes you can get weekend specials at chains like Thistle, Novotel, Jurys or even Holiday Inns or Hiltons if you get lucky.
In my experience looking for hotels on the internet, Bed and Breakfasts are often not a lot cheaper than a 2 or 3 star hotel or inn and indeed, many B&Bs are actually small hotels. Some will have shared facilities and some with en suite. Shared facilities are cheaper but for me, I prefer to be the only one using the toilet and shower during my stay. Many hotels have at least a continental breakfast included in the rate which is fine for me. I don't need a heavy cooked breakfast every morning. If you have to buy your breakfast, avoid the hotel since hotel prices are usually far higher than need be. Find a nearby bakery or cafe instead.
I've stayed in a few different hotels in London. In general, I don't pay more than about 75 pounds per night unless I'm splitting the cost with someone. With the Canadian exchange rate being so low at the moment (2010), we can get a 100 pound hotel for the same price in Canadian dollars as a 75 pound hotel used to cost so that's always nice. If you're paying less than about 60 pounds for a hotel, chances are it's very, very basic, or way out of the centre of the city or it's a dump. Even the 60 pound range is pretty iffy these days. For a decent hotel in a decent area close to transportation and with en suite facilities, you will likely pay about 70 to 75 pounds a night unless you manage to get a deal. Lots of people get lucky with hotel auction sites like Priceline or with "hidden" hotels (i.e. you won't know what the name of the hotel is until after you've paid for it) on travel sites like Hotwire.com or Travelocity.com. You can save money with those sites though you won't have a choice of the hotel. I think they mainly use decent places, though, and in the popular areas so taking a chance could very well bag you a really good hotel for an affordable price. Your mileage may vary. I like to know what I'm getting first.
The two booking sites I've had the best luck with are LondonNights and from the official London tourist site, Londontown which is also a good site to get discounted theatre tickets. Both have a variety of hotels from budget to luxury in many different areas of London. Neither site requires you to prepay the hotel stay.
Of the hotels I've stayed in, there are a few I'd definitely return to. I've included individual reviews for Strand Palace Hotel, Hilton Islington, London Guards hotel and Thistle Euston in this journal. The Hilton is usually out of my budget but I'd got a good deal (65 pounds with full buffet breakfast for two!) through Londontown.com on a weekend stay. The Strand Palace is a bit more expensive than I usually pay but I was splitting the cost with my mother and I had wanted a good, central hotel close to bus transportation for her first visit to the city since she wouldn't use the underground. (too many steps and too claustrophobic for her) Another tourist class hotel I used some years ago is the Tavistock Hotel. I haven't been there since 1993 so I have no idea if it's still ok but I think it probably is. It will be a bit older, possibly a little worn around the edges but I recall that it was clean and near two underground stations (Euston and Russell Square) and a short walk from the British Museum.
As always, your mileage may vary but these have been my most positive experiences.