What sites are recommended for a 4 year-old in central Stockholm? Where are the best parks?
If he/she likes ships the Vasa museum would be nice
Skansen and the Vasa are musts. They are both on the same island but with a 4 year old you might want to split the visits into separate days. The child might also really enjoy the harborfront - lots to see with boats, birds and such. There is a daily (I think) horse parade to the Royal Palace and daily cannon salute off one of the islands (the kid will probably be thrilled to get the earplugs if you are on the island with the cannon). Check the guidebooks for times on those.
One very popular place for young kids in Stockholm is Junibacken. It is a children's museum featuring the stories of Astrid Lindgren. The kids can ride a train, hear stories (ask for English) and visit Pippi's house. It is by the Vasa Museum on Djurgården. Website is www.junibacken.se
Depending on when you are going, Skansen has children's events. In particular, there is a zoo in Skansen with Scandinavian animals; I've visited there with friends with children and they really enjoyed the animals. The Skansen website lists events of interest for children. http://www.skansen.se/kategori/kul-barn
The park called Humlegården is a nice green park with a children's play area. You might consider getting food for a picnic from the indoor food market Östermalms Saluhall (www.saluhallen.com) and going over to Humlegården. There are other public parks with children's play areas. Ask at your hotel.
One more thing ... while you are there (or perhaps before you visit) look for the children's books by Sven Nordqvist with the characters Pettson and Findus. These are wonderfully illustrated books with very clever stories. You can find them in English.
Thanks for all the advice! We are very excited about going. We're currently looking for an apartment of house to rent. No hotels for us this trip.