Honolulu is a very family-friendly destination.  Ohana (family) is important to the Hawaiian culture.  The obvious choice is the world-famous Waikiki Beach.  This is a very family-friendly zone.  There are many hotels that cater to children and have numerous activites to entertain them while visiting the Island.  The Hilton Hawaiin Village has dolphin-petting, which is a favorite with small children.  A small list of activites for your family to enjoy while visiting Waikiki include:

The Honolulu Zoo - www.honoluluzoo.org- if you are a member of another zoo, check to see if there are reciprocal discounts here.

The Hawaiian Water Adventure Park - www.hawaiianwaters.com- the island's biggest and best water park, it's a bit out of the way but worth it.

The Polynesian Cultural Center - www.polynesia.com - a great day trip, with several "islands" offering tastes and sounds of their own local culture. Crafts, games, and dancing make this a wonderful stop for any family.

The Waikiki Aquarium - www.waquarium.org

Whale/dolphin watching cruise - several operators to choose from, but this is a great evening activity. 

There are also many shops and boutiques to choose from, snorkeling excursions and much, much more.  Honolulu is a fantastic family destination!