For kids

http://www.gibbonproject.org/content/about_us.htm
hire a car, it's easy to drive in Phuket

The elephants rides of course are attractive because every child loves elephants - however, they are really touristy & not natural at all. It's not trekking. The elephants are chained up. Be prepared for that.

Sea kayaking, if your children swim.

http://www.splashjunglewaterpark.com/ (that is up in the north of the island, actually just up the beach from the Holiday Inn Mai Khao, but there are not really dining out options nearby)

Go fishing? My mum loves that

The Phuket Aquarium is not bad and very good value. http://www.dmcr.go.th/phuketaquarium/?lang=en

Do they like cashew nuts? You can visit a (commercial) cashew nut factory to see how they are produced & try many flavours. A good place for half an hour if you are travelling between activities.