Greenwich - can combine with Ahoy kids play area at the Maritime museum (need book now I think) but enough to do just wondering around town centre/market/river/park/playground
Greenwich peninsular area. Great scooter riding down by the river walk, can go on the cable car and pop into o2 to marvel at its strangeness. Also a mini art gallery and random sculptures dotted about
Canary Wharf - can combine with mudlarks at docklands but again fun just to mooch around and look at the water and crazy buildings. Shopping centre, food court, little parks.
Tower bridge/Bermondsey area and cross over the river for a mooch around the marina
Elephant park - especially when it gets hotter for the best water play
Peckham rye - 2 playgrounds, nice cafe, lots to explore
Deptford market - buy random things for your house for 25p, see a show in the Albany theatre go to the deptford library to read books (the library is actually v nice)
Hillyfields park for a cute cafe, playground and nice views