This is one of the things your children will no doubt insist upon once they find out it is possible to ride an elephant. For the adults that accompany them we have some words of advice: Those wooden seats are not comfortable. Choose a trek that covers flat ground, if possible, and keep the trip to 30 minutes to an hour. The children will be happy, and you won't be too sore.
The Elephant trekking companies on Koh Phangan are particularly nice to the adults in the crowd, because they offer a service that doesn't involve sitting on the elephants. The children can feed bananas to the elephants, which may satisfy them without requiring a long ride.