Go to Oxford Street and go shopping. And Hamleys on Regent Street, you could kill a few hours in there!! Go to Harrods and have a look, went there with a friend a couple of weeks ago and we managed to spend 3 hours in there just looking, they have amazing things!!
Walk around Covent Garden and look at the street entertainers.
Go to the museums/ galleries, a lot of them are free and a nice way to spend a couple of hours.
Markets- Camden market is good for clothes and random bits. Borough market is really nice for some good food (one stall has a brownie tower, it's huge and super yummy!). Portobello market is apparantly nice though I haven't been there.
Wander round Westminster- you can see the Houses of Parliament, Big Ben, Westminster Cathedral, Downing St, Trafalger Sqaure, Horseguards Parade, walk through St James's Park to Buckingham Palace. Photo opportunities galore!!
Even if you are on your own you'll have a great time, there's so much to do in London you won't be bored.