While I'd agree that Bath and Stonehenge are definitely "must sees," I wouldn't burn a full day on the trip with only a weekend in London. There are tons of things to do and see within the limits of London, regardless of where you stay. You can pick up a 3 day travelcard (or 2 one day cards, depending on your definition of "weekend") to get around on all means of transportation and easily hop on and off to get to just about anything. See http://www.tfl.gov.uk/tickets/faresandtickets/oneand3daytickets/2911.aspx for more info.
As to what yo do, that depends on what you like. My wife and I enjoy theater, and as such usually hit Leicester Square for half price tickets at least two to three times a week when we're in London. But we've done all the "touristy" stuff like the London Eye, Tower of London, Tower Bridge, Westminster Abbey, Royal Botanical Gardens, the Dali Museum, St. Paul's Cathedral, the British Museum, Tate Modern, City Museum, the Victoria and Albert, and so on.
There's a London flavored version of Time Out magazine (I think they may actually be the original) which provides weekly guides to what's going on - art, music, theater, nighclubs, sports, etc. You can pick up a copy at any newsstand.
If you're in to shopping, Kensington High Street is the place to go. Harrod's is certainly overrated, but many consider it to be the must-do in London. Keep in mind it will be very crowded on a weekend and thus any plans for high tea would be better had at a smaller establishment if you feel that's a must do. (If you're not jung up on the doilies, you can get excellent scones almost anywhere in London save for Starbucks.)
There's also tons of free stuff to do if you are interested. Time Out will list many options, but there are some good resources you can check out ahead of time as well: http://www.londontourist.org/free.html and http://www.allinlondon.co.uk/life/index.php?fid=5.
One last plug for Time Out magazine (and no, I don't work for them, though I did live by them when I was in London in my college years): they have a list of 101 things to do in London before you leave: http://www.timeout.com/london/features/2253.html Obviously, it's a bit much for a weekend, but maybe it will give you some ideas.
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