We have traveled to London 3 times with our children, first when they were 3 and 6 and more recently when they were 10 and 12 and again when they were 11 and 13. We have stayed in hotels and rented apartments, taken taxis, buses, and the tube, and we have eaten in a lot of restaurants. Here are some tips that might be helpful to others traveling to London with children.
1. Rent an apartment. You can find great 2-bedroom apartments in good parts of town for $250-$300/night (and even better prices if you stay for a full week). Decent hotels will be this price for a single room. Granted, you don't get concierge or housekeeping service, but you get space to spread out (essential with pre-teens and teens), a kitchen to prepare some meals (and save some money), and often a washer (and dryer) to do some laundry so you can pack light.
We stayed in Soho one year. It was convenient, but very noisy and crowded. This year, we stayed in Chelsea. I highly recommend any area of Kensington, if you can find something in your price range. It is convenient to central London, near museums, full of shopping and restaurants.
2. Don't be afraid of chain restaurants, but pick good local chains. London is actually a great place to take kids out for a meal--many restaurants have kids menus and the service for kids is great. Chains we depend on are Pizza Express (really very good pizza), Giraffe (all over Brittan), and Carluccios. This year, we also went to Maxwell's, which was great for comfort--and the food was much better than similar "American" chains such as Planet Hollywood or Rainforest Cafe.
3. Look for unique restaurants in your neighborhood or where you are going to spend the day. By mixing it up a bit between chain, comfort food places and more interesting places, we all were happier.
4. Consider where you will be visiting before you buy a London Pass. We skipped this because we weren't going to do many of the things you get with it. With kids, we found we were going to the museums (free) and that any paid attractions we chose to do (such as Legoland) were not enough of a deal to make the London Pass worth it. If your kids are interested in seeing Westminster Abby, St Pauls, and the Tower of London, then it makes sense. Our kids just wanted to be at the Science Museum, Victoria and Albert, and the British Museum--all free.
5. Take the tube and buy travel cards. Adults can buy Oyster cards (easy) and put 7-day travel cards or "pay as you go", which is cheaper than buying a ticket each time. You can only buy kids' travel cards one day at a time, so you'll need to buy them every day. But, you can buy them from a machine (which we only figured out part way into our trip). AND kids under 11 are free! The Tube is fast, easy, and fun--we usually got places faster than if we had taken a cab.
6. If you are going to be in London for many days (5 or more), consider going to a movie in the middle of your trip. There is something really soothing about doing something as normal as seeing a movie, and doing it in London--where it's called the cinema and the ads before the film are British--is fun.
7. See a show in the West End. We've seen Wicked, Spamalot, and Mary Poppins--and we've loved every one. During the first weeks of August (which are generally very crowded in London because all the kids are out of school) you can get a free kid's ticket for every adult ticket you buy to many shows. Look for them in advance, though, because they go fast.
8. Don't try to do too much in one day. This is another reason why having an apartment is great. You can sleep in, take a break in the afternoon, take a day off from sight seeing, cook dinner at "home".
I hope you have a great time. London is a great place to travel with kids.