What is the most cost effective way to stay in London for a week? Should I stay at a B&B, rent an apartment for a week, or stay at a hotel? Should I stay in London or the outskirts of the city?
Regarding cost effectiveness, it will be difficult to beat Couchsurfing. Where to stay depends on what you want to do in London ... Try to stay near the area you want to visit, otherwise you will incur travel costs.
The most cost effective will be a youth hostel. You can find them just by doing a research on Google or on travel website such as Expedia, Hotels.com, etc. The outskirts of the city will be less expensive than the center of London but check on TFL for the travel card price before taking a "far away from the center" place to stay !
This really depends if you want to spend the least money possible (sacrificing some things on the way) or if you have a minimum level of comfort / ease of travel / etc you .
The current answers are very good low-cost options, I agree that Couchsurfing is probably your cheapest option, but not everyone wants to do that.
If you want a hostel / hotel then go to a hotel comparison site, Raiana linked some but I tend to use Kayak. I stayed in London a few weeks ago and booked a hotel near Paddington that was only a tiny bit more expensive than the nearby hostels -- plus I got a private room with en-suite and free breakfast. So definitely look at your options, there's no hard and fast answer.
As for where to stay, consider how you're getting in and getting out of London. Do you need to be near an airport link due to an early flight. If you stay in the outskirts travel will cost more (and you'll need to leave town earlier), plus you're looking at possibly an hour plus on the train in each direction.
Without knowing your budget, how many of you there are and what you want to do it's difficult to say more. I'll edit this answer if you provide more information.
A few other money saving tips -- if you want the cheapest travel consider getting an Oyster Card, but it depends what you want to do. If you plan on a lot of sightseeing consider a London Pass (possibly with the travel card option) but again that only makes sense if you plan to see enough to make it worth it.