As a Britischer, I am often asked for suggestions by people who plan on visiting London: where to eat, what they really must see, and where there are good hotels. For the latter question, I always use Trivago (UK/CH) when searching for hotels, as their site collates information from all of the other price comparison sites and so you get a better overview.
Specifically for the U.K. and when price is a main selling point, I usually recommend Premier Inn as a hotel choice. Sure, it’s a chain hotel, and many people would prefer to stay in a trendier place. But Premier Inn rooms are of reasonable size – even in central London – and of a good standard across all of their national locations. You’ll have enough things to spend your money on in London, without forking out too much on a shabby little room which you’ll (hopefully) only see when you’re asleep.
Premier Inn have a Good Night Guarantee if your stay isn’t satisfactory, free wifi, and their hotels are usually in very convenient locations. In London, I’d say that Southwark, near Tate Modern, is the best option for its combination of price and location, whilst County Hall is in a superb location, right next to the London Eye. Don’t be put off by the big railway line near the Southwark hotel: the rooms are well-soundproofed and the railway isn’t right next to the hotel.