I'm not a mum yet (got a few months to go!) but I am a frequent London visitor who often takes nieces and nephews. In my experience, it is best to go on a Sunday crowdwise. It's still going to be busier both in the city and on the trains than it would be on a weekday, but Saturday will almost always be the busiest of the week fullstop.
Get a railcard (you can often make the cost back within one or two trips depending on where you go and how often), and look out for Summer advertisements for cheap group rates (there's almost always something on).
You probably already know about some offers, but Tesco Clubcard do a 2 for 1 entry on a lot of places, I think the vouchers cover the likes of London Aquarium and the London Eye the last time I went.