You have to do a museum, but which one? Tate Britain for the most marvellous JWW Turner pictures in the Clore Gallery, Tate Modern at the South Bank for its regular collection (free!), National Portrait Gallery for everything, but especially the annual portrait competition, Victoria and Albert for the costumes and musical instruments collection, Natural History museum for the kids, and there are TONS of little museums as well.
Hottest ticket that you won't be able to get without booking a year in advance? The Whitechapel Bell Foundry is a business that created Big Ben and The Liberty Bell. You can join a tour on Saturday mornings, but you have to book WELL in advance. They are sold out through 2009 now. Great two hours on a Saturday morning.