For families with kids under 5 (and over 5 too!):
1. Tate Modern - the building is fantastic (dramatic; it captivated our daughter when she was 18-24 months), the art stimulates curiosity, and views of the City (St Paul’s, etc.) are amazing. Excellent kids' books in shop.
2. Tate Britain. Get the kids’ activity pack or the free children's audio guides. Despite the name, this is nowhere near the Tate Modern and the collection is very different. You can easily get between them however on the Tate-to-Tate Boat which runs along the Thames, past the Eye, and takes 18 minutes each way. This was the highlight for my 3-year-old, and we went back and forth an extra time for a total of three 18-minute journeys (get the day pass for less money than two singles!)
3. Science Museum -- don't miss the area for younger children buried in the basement, with a water shooting table, magic mirrors, etc.
4. Natural History Museum -- dinosaurs are the favourites with kids I know. Special show on 'Hair' (just hair, not the musical) in summer ‘04.
5. V&A across the street has activity carts for kids. HUGE!
Quick Tips:
All these museums are free! Always get Time Out magazine when in London for a few days, and check for special events. Many museums have shows or talks or crafts for kids -- ask when you arrive.
There is a grassy area outside the Natural History Museum with ice cream trucks. A 10-minute walk north to Hyde Park will bring you to a playground.
Natural History Museum and Science Museum have tasty cafes with mix and match lunch specials for kids.
Major baby/child tip: Ask where is the Baby Care Room (children's toilets) in the Natural History Museum; just to the right off the large entry foyer.
Best Way To Get Around:
Combine Tate Britain and Tate Modern by taking the Tate-to-Tate boat between them. Discount if you have a travelcard -- i.e., 3.40 for as many trips as you like all day.
3, 4, and 5 above are all next door to each other, though all so huge you'd have to be very selective to do all three in one day. Arrive at South Kensington station. Walk outside if nice weather, or in dank but covered tunnel if it’s raining.