London with a 13 year old boy

The HMS Belfast, a WWII destroyer that is now a floating museum, is berthed on the Thames near city hall. It takes about a half day to thoroughly explore this ship but you can do it in less time if that is a constraint. On the more obscure side, there are about a half dozen army regimental museums located in London starting with the one located at the Tower of London. I have never visited one of these so I cannot tell you if there is much pertaining to WWII.

Please note that the Imperial War Museum will be closed for a major renovation "for 6 months Jan-July" (as per their website). We just found this out, a major bummer as that is on our list, we will be there July, and one week of our lodging is across the street from it.
The IWM does have an additional location in Duxford, south of Cambridge. It is primarily a huge aviation display with over 200 planes.

Army Museum, Horse Guards Museum, Science Museum, Natural History Museum, zoo, aquarium.

Two sites that you may like to consider the Imperial War Museum in Lambeth just south of the river and the War Rooms just off Whitehall near ST James's Park, very close to the Palace of Westminster. There is also the Army Museum in Chelsea and the RAF Museum in Hendon North London.

There is also a tank museum that is between London and Weymouth. It will take several hours by train to get there: http://www.tankmuseum.org/ Several times a year they have a class and let civilians drive a tank. It's pricy: in the neighborhood of $500 USD. Might be worth it if they let you fire a few rounds.

Am a WWII buff and took my teenage son to St. Paul's Cathedral to the back to see American Chapel dedicated to US servicemen stationed in UK who died during war. Interesting story about why and how it was dedicated, can find info on the St. Paul's website.

Tower of London- the Beefeater guiedes have some gruesome stories to tell

Agree with the above re London Walks. Walks.com. I have done their walk that ends at the Churchill War Rooms. I also did their Explorer Day to Bletchley Park - but this is easily done on your own, a short train ride out of Euston. Another possibility to add to your considerations would be Chartwell, just out of London, which was the home of Winston Churchill. Check their website - its a National Trust property. I went there a couple of years ago and it was much more interesting than I expected (and I'm not a WW11 buff). It has the house as it would have been when Churchill lived there, + several rooms given over to an excellently laid out museum and really pleasant gardens and grounds.

Thanks to all who have taken the time to reply. Looks like we'll be needing more than 4 days for our visit! Is there any way to circumvent the long lines at some of the major sites, eg. Tower of London, as we will be visiting in August.

You can buy Tower of London tickets online before you go. There is a place to pick them up when you arrive. It is to the right as you enter the area. I bought the tickets on my phone the day before. It saved a lot of time. I think there is info in the RS book how to do that. At St. Paul's you can climb all the way to the top. It gets pretty narrow before you get there! We loved it especially our teen-age son. The views from up there are great.
We also enjoyed going to Greenwich with the Prime Meridian and all the time keeping history and a great maritime museum.

Don't forget the British Museum! Haven't met a boy yet who didn't like the mummies.

If you can, get to the Tower of London first thing in the morning, so you're there when they open. I've done that and in my opinion it made the visit more enjoyable. We didn't have the place all to ourselves, but it wasn't as full as it got an hour or so later. By the time we left, I was glad we were leaving and not arriving. (I enjoyed it all much more than I expected, and now that it's been a few years, I'm entertaining the idea that when I go back to London I want to visit the Tower again.) Another vote for the Cabinet War Rooms. Low-key, the real thing, and I found the place so fascinating I've been there twice now.