Take The Tube
The TubeOK, you'll be lucky if you ever find a carriage this empty. But the tube is without a doubt a defining charm of London (even if it's the bane of the lives of those forced to commute on it during rush hour). If you're unsure where you want to visit, or you're bored of ticking off sights, a ride on the London Underground is a great way to find yourself in new and unexpected parts of the city.
Take a look at the map: Anywhere bounded by the Circle Line (the yellow one) is central enough to have something exciting going for it. Hop on, hop off—you're never far from the action.