I strongly suggest you to try websites like Spareroom, and to stay away from Gumtree where a big percentage of the ads are scams. Every room I found in London was through Spareroom, you can pay a small fee to have early access to the newest ads and it seems to be worth it.
East London, like everyone else said, is a great place to be, but it’s getting harder and harder to find a decently priced room in the area. You can get away with paying 500 ~ 600£, but you’ll have to be willing to share a house with 3 or 4 other people… it can go very well, but it also can go very wrong.
If you stay around Hackney/Brick Lane, I can recommend the co-working space my company uses, called East One Studios. Not expensive and quite cheerful the benefit of this area is also having places like Google Campus relatively near. Free wifi and working areas, although they can get quite busy (it’s London after all).
It’s definitely a challenge to save money in London, but after a while you learn the tricks. I’d strongly recommend to be in cycling distance from whatever working space you choose to be in. This is the best way to save money on transportation, which is a good chunk of money per month.