One of the many free events my program provides is going to the Transport Museum. It is among the few museums in London that actually charges for entrance...and truthfully I am not sure why. It was a great time and very interesting but my friend and I looked at the whole museum in about an hour.
It was very interactive and catered a lot towards families with small children. If that is your situation then I would highly suggest visiting! It was awesome to see how excited little kids were getting learning about the history of transportation in London!
If you do not have young kids then I would see if there are any coupons for discounted admission and if not I would personally just stick with going to the free museums like the National Gallery and Tate Modern (where you can spend hours getting lost).