South-West London:
Kew Gardens already mentioned, but excellent for running around, and for little ones Climbers & Creepers is fun - they also have art activities in the holidays
Wetlands Centre in Barnes - new playground opening this weekend! Also pond dipping, indoor children's displays, nature trails and lots of walks. Oh, and you can even look at some birds.
Richmond Park - for cycling (hire bikes and trailers near Roehampton Gate), small playground for under 5s near Kingston Gate, lovely walks in Isabella Plantation and Pembroke Lodge Gardens, great climbing trees near Pen Ponds
Hampton Open Air Swimming Pool and Tooting Lido
Pools on the Park in Richmond has an open air pool too, but it's heaving in summer and the others are nicer
Drop in tennis and football every morning at Rocks Lane Tennis Centre in Barnes - you can buy tokens and just turn up when you want to (sorry, don't have the number to hand)
Kew Bridge Steam Museum - good for train-obsessed small boys
Syon Park - expensive, but lots to do: Butterfly House, Reptile World, a good garden centre with pet shop, the infamous Snakes & Ladders indoor play centre, Syon House and gardens itself which has a train ride in summer
Walk along the towpath from Putney to Richmond and beyond - any section really. We tend to do from Mortlake to Hammersmith or Mortlake to Richmond, coz of where we live, but there are lots of things to see and do on the way.
Hampton Court - haven't been for a while, but maze, gardens and house all good
Bushy Park - big sandpit for little ones
Battersea Park - zoo and adventure playground. Had a good day out there at half term.
There is a programme of outdoor plays on in Richmond's parks, but I've forgotten what it's called and have lost the leaflet.
Will post more later if think of anything else locally.