The National Folk Museum inside the Gyengbokgung palace looks like a great place to take kids of all ages, including toddlers. According to this summary from from the Visit Seoul website, there are playground-type activities that incorporate the folk and natural history of Seoul, interactive Korean culture activities and other fun things to touch and play with. One thing to note is that reservations for tour guides are recommended.
Here's what visitseoul.net says:
"The Children’s Museum is an interactive children's educational space attached to the National Folk Museum inside Gyeongbokgung (palace). The first floor holds the permanent exhibition and the second floor is used for special exhibitions. The permanent exhibition uses the familiar folktale of Simcheong to depict traditional Korean daily life and culture while the second-floor exhibition rotates every year. It is a good idea to make advance reservations so the staff can guide you through the sights and educational experiences."