With children that young I think it would be less about which island and more about where you can stay that will have fun stuff for the children to do.
I've been to Kaua'i and Big Island, and spent a couple of hours on O'ahu.
All of these islands have nice beaches, shops and tourist areas.
Kaua'i has a gentler, slightly cooler climate with a tendency to afternoon rainstorms. The worlds wettest spot is here - the highest point has a mist/gentle rain that almost never stops. Seeing two rainbows at this spot is almost a given. The waterfall from Fantasy Island is also on Kaua'i. There is a building ordinance here that requires all structures to be no taller than a palm tree - this gives the island a more rural feel.
Big Island has Volcanoes national park and Mauna Loa. I think both places are way cool, but I don't think that toddlers would be much interested. There is a nice small protected beach in Kailua-Kona where giant turtles and all sorts of tropical fish are found. They're very used to people. (I've had several glare at me because I didn't have a snack for them) Big Island also has several large resorts on the dry side of the island - these are very self-contained. This part of the island is very hot and dry.
In Oahu, most people stay in Honolulu. Honolulu is a very urban large city, with skyscraper hotels all the way to the beach. I thought it was very crowded.
Maui is of the party/resort tendency with a lot of beach amenities; I've never been there, but my brother and SIL have.
The best time to go is probably Summer - in the winter the seas are rougher which makes the water cloudy and snorkeling unpleasant. I've even been on Big Island when there was a cold front come through and it was a bit chilly.