I love London and spent weeks there as a teenager enjoying all the museums. I took my kids there for the first time (they are 9 and 12) and after a week of touring around, here are some impressions:
London Eye - Awesome! Stunning views from what is essentially a 400 foot Ferris wheel with enclosed cars. Make sure to make a reservation in advance to save time
Science Museum - Great interactive area for kids in the basement. Fabulous exhibits - I liked Babbage's difference engine in particular. And its free
Somerset House - The best museum you never heard of -- fabulous collection of Impressionist paintings that I thought was better than the national gallery, and, in another area, a wonderful collection of the most amazing stone mosaic work you will ever see
Theater - my kids love Broadway shows. We saw the musicals Mary Poppins and Mamma Mia and the play Mousetrap and all three were great, though expensive. There is a TKTS office in Leicester Square that sells same day discounted tickets.
Imperial War Museum - Tanks and Missiles, what more could you want? The collection is huge, and most kids will tire of the uniforms and such, but its free and worth a quick visit.
St. Pauls: Just spectacular. The downside is that the admission is expensive, but I found it worth it just to stand under the dome in awe.
Cabinet war rooms: The underground chamber used as a senior command post in WWII, it sits (with papers, maps and all) just as it was in the final days of WWII.
Tower of London: Great, even before you see the jewels. However, take the beefeater tour. It is free and you will miss 90% of the experience without it.
Harrods: My kids loved seeing this unique store, particularly the food courts
Hamlys: 6 floors of toys. Say no more.
Buckingham Palace, Windsor Castle, Hampton Court. Day trips to the latter two are recommended, particularly to Windsor Castle. Buckingham Palace is only open for tours a few months of the year, so we missed it, but I toured it a few years ago when they opened it after the Windsor Castle fire and it is beyond amazing.
Maybe / Maybe Not
British Museum - I love it. This may be the greatest museum of archaeological finds in the world. However kids, after some initial enthusiasm for the mummies, tire quickly. Its just so big
Museum of London. Again, I loved it. My wife and kids were bored. If you are the type that likes to read all the little cards in museums, this one is a great history lesson. If you like to breeze through and look at the cool stuff, there is not as much here for you.
Westminster Abby - If you go, take the tour. Its expensive, but it is not as architecturally interesting as St. Paul's. The interesting stuff is in the history of the tombs and who is buried there, and there are very few signs explaining what you are seeing, so you will miss most of it without a tour.
Tate Modern - We did not go to this relatively new museum but friends of ours took their young kids and said it was great
Double Decker Bus Tour: There are two major companies that operate these, and you get a day pass so that you can get on and off the bus all day. The tour was pretty good, but we found by the end of the week it mostly covered ground we retraced later in the week on our own.
Royal Mews: The Queen's stables and carriages. This was OK, but not great.
Not worth it
Madame Tussuad's: Expensive, ridiculously crowded, and pretty short (we took our time and were done in about 30 minutes). If you go, make a reservation in advance or you will be stuck in a very very long admission line. My kids thought is was OK, if only to have a picture of themselves with James Bond and Saddam Hussein
Natural History Museum: Despite its incredibly rich history, we found this museum disappointing. The entry is beautiful, and the museum is free, so its worth just checking out briefly. We, however, found the layout to be awkward and not very interesting. The post-renovation natural history museum in New York I think is both more entertaining and laid out better to really teach you something about nature and evolution.
National Gallery: I have given this museum several chances, and I find it disappointing every time I go. It doesn't stand up in my mind to galleries in other cities like Paris, NY or even Chicago. Its free, but I don't think it will satisfy either art lover or non art lover. Go to Somerset House instead.
The underground: Wow, has the tube gotten expensive! Three pounds per person one way. This meant for our group, it was less expensive to take a taxi. We tried to walk as much as possible, took taxis when we could go no further, and took the tube only once or twice. The tube may work better financially if you buy a weekly pass - we did not look into this.
Hot Weather: It was in the high 80's and low 90's when we were there. Note that much of London is not air conditioned and the rest is inadequately air conditioned. We roasted in the museums and in the theaters. If it is hot, get used to hot, uncirculated air in crowded places.
We found a pretty good hotel for families called the Ascot-Mayfair. It has a great location near Hyde Park Corner and has multi-room suites for decent rates (at least for London). The one bedroom suite has a sofa bed in the living room that will sleep the kids and get them out of the parents room. We splurged on the 2 bedroom suite, and got it for a rate less than the tiny one room place we had in Paris last summer.
I posted earlier about the strange bias against kids out in the English countryside. We saw little of this in London. A few hotels would not let us rent rooms if we had children, but otherwise no problems. Restaurants were all very accommodating, many with children's menus (which is a big change over a decade ago). Also, restaurants and shows are quite informal so that we had a lot of nice dinners without taking any really nice clothes. For restaurants, we particularly liked Ping Pong (Trendy Chinese dim sum), Yauatcha (also trendy chinese dim sum) and Lucio (Italian).