Nat history museum Science musuem esp downstairs in the basement London Eye Duck tour or one of the boats down the river Open top bus All the big parks have great playgrounds Zoo (combine with Regents park for playground and rowing boat) Good restaurants on the south bank eg Giraffe
They could easily fill a weekend down by the river if they're in Southwark.
Wagamama They are all over the place. They do children's chopsticks, which make DD feel very grown up, and the ice lollies are to die for.
The Princess Diana Playground in Kensington Gardens. Not only is it free but is a (longish) walk from the Science/Natural History Museums. DD loves the V&A which is across the road from the Natural history Museum. It is never as crowded, does kids activities and has a pond in the courtyard which is always full of paddling youngsters on warm days. (and it's free).
If it gets cold, take the DLR (sit in the front seat and 'Drive' the train) to Stratford for Discover. Very cheap and always a hit with the small ones. The Jubilee line goes there too, but be warned, if you are going on a weekend, the line is often closed for repairs and upgrading.
If your DD is an adventurous eater, try either Yo Sushi or an independant Sushi bar. There is one in Greenwich with a travelling belt. DD is endlessly fascinated by it and loves Endame beans and Miso soup.
I second the Princess Di playground. Good for under 5s. Bring a change of clothes if your little one likes sand/water, though. If it's sunny, bring a cossie and towel as well, as lots of small kids like to go splashing in the Princess Di memorial fountain. There's a nice old fashioned carousel there, too, and the Serpentine and cafe are only a short walk.
5 min walk from the playgound to Whiteleys shopping centre, which has an Ask and a Cafe Rouge, and there are loads of Chinese / Dim Sum places in Bayswater
A wander around Borough Market will result in the child being fed for free from all the samples set out on the stalls. My DD who, to be fair, is an absolute glutton with a very sophisticated palate, refers to Borough Market as "The place of scoff" It gets very crowded on a Saturday though.
Gaby's, opposite Lecister Square, on Charing Cross Road, just down from the tube station here is very cheap and cheerful for such a central London spot. They do fabulous salads and hummous etc.