I see London, I see France, I see... OK never mind.
Well, I'm not in either London OR France, but I'm dredging stuff up from WAY back when, when I did a work exchange in London. (Good lord, that was almost 15 years ago, now!) So keep in mind this is all out of date.
I really loved the Tower of London. Yes, it's touristy, but there's a reason for that... because it's awesome.
![Smile :-)](./images/smilies/icon_smile.gif)
Ravens AND weapons AND a castle! I suggest either skipping the Crown Jewels or going first thing in the morning or last thing before closing, when WAY back when there wasn't usually much of a line. They're shiny and all, but not worth waiting in a huge line for.
The British Museum is fantastic.
I don't recall being that bowled over by the Natural History museum, but then I was lucky enough to grow up in a city that had a very good natural history museum that they've kept very well funded, so it takes a lot to impress me, natural-history-museum-wise. And I don't know how much has been done on it since then.
Depending on your interests, there's the Royal Observatory in Greenwich or the British Library (the former for Observatory-related stuff and Harrison's chronometers, the latter for old books, letters, illuminated manuscripts, etc.) If those are the sort of things that sound interesting, you'll probably find them interesting. If not, you won't.
I recommend against The London Dungeon (tacky and not actually informative) or the Toy Museum (heavily advertised, but boring, or at least it was then).