Hello Notalone
You probably want to avoid doing too much as you mention four places but they are not very close together! For example, on the first day if you come into (I'm guessing here) Kings Cross then I would allow an hour to get to the Science Museum (Piccadilly line to South Kensington). After the Science Museum you could nip next door to the Natural History Museum to see a dinosaur or two. Apparently they also have an exhibit of live butterflies at the mo: kids.visitlondon.com/whatson/detail/2433432
Some people would also stroll down the road to Harrods- it's not really my thing but a small person might be impressed by the food halls for example.
You could then catch the District Line to St James' Park (as an alternative to Hyde Park), see a pelican or two and be nice and close to Victoria station.
On the second day, can I suggest a bus trip? This bus map is quite helpful. Your DS might like sitting on the top deck at the front. The Number 11 bus route is particularly fun, it goes from Victoria to Trafalgar Square past Big Ben and Downing Street.
www.tfl.gov.uk/assets/downloads/central_bus_map.pdf
If it rains, have you considered the London Eye? I know it is expensive but it really is a great way to see London from above. You could combine it with your stroll along the South Bank, from Waterloo station. If you have time, my favourite walk from Waterloo involves heading right (ie. East) and walking to the Tate Modern and back.
Can I ask, will you have luggage? Have you thought about where you are going to leave that? Victoria does have a left luggage place but it is expensive and you would have to go out of your way to drop stuff off/ collect it, which would also constrict the amount of things you could see. Things can be quite spread about in London and I wouldn't want you rushing about and not seeing things, or worse, missing your connections. It is probably realistic to concentrate on the West of London (ie. the museums) on one day, and the centre (ie. the South Bank, Trafalgar Square) on the other.
And yes, tubes and buses are free for children. You can go to this website www.tfl.gov.uk and it will tell you how to get between any two points in London.
I am quite excited for you both and hope you have a great time!