Well, I'm in greater London and don't go into the centre all that often, but the Docklands Light Railway is fun, as is the cable car over the Thames at Greenwich.
Strolling along the South Bank is good but can be a bit busy. I like Canary Warf area on a weekend as it is calmer.
The parks are good, spacious and often have amusing things to watch i.e. roller bladers, Hyde Park ranters, boating lake, ducks, trees to climb.
Changing of the Guard? Just looking at them is quite entertaining. Waiting for Big Ben to Chime.
I like Coram Fields too.
Tate is fun for it's big slope to run up and down.
There are LOADS of museums, and most are massive so concentrate on just one and just one section of one to avoid overload.
I took DS to the Royal Albert hall recently and though expensive was imo the best money I had EVER spent. From the second he entered and announced 'gosh what a lot of people' until the end, was utterly utterly magical.
There are some posh adventure playgrounds in Kensington iirc.
Walthamstow Market is a mile long and still quite tradition with the vendors shouting out their adverts.
There's the gerkin with spirally ramps/paths.
There's St Pauls with the whispering gallery, or just stand outside and sing 'Feed The Birds'.
Riding the buses can be fun, especially the number 4 who they appear to deliberately recruit crazy drivers who go a different route every time.
St Martins often has free lunch time music. Trafalgar square is good for people-watching.
Embankment Gardens is nice for a stroll, and there are plenty of boat rides on the Thames.