I'm in Oz so I'll let you know about some options down this way.
I wouldn't recommend New Zealand or most of Australia during the Northern Hemisphere summer unless you want to ski. It's cold! Far North Queensland would be pleasant though.
Destinations that occur to me from down this end of the world are Pacific Islands or South-East Asia.
If you want to cruise, a northern Great Barrier Reef cruise could be idyllic, and no stingers around at that time of year. If money is no object I believe Orion does a cruise from Cairns and across the Spice Islands that looks fascinating. Cruises across the Pacific would be good if you've got the stomach for ocean crossings, or you can do shorter cruises just around French Polynesia, or hire a bareboat and sail around Tonga.
For a land-based holiday, there are three options I'd recommend, depending on your budget.
Your best budget option is Thailand. If you go for one of the less well known spots such as Koh Lanta, you can get fantastic bargains. eg the small family-run beach-front resort we stayed in there cost $36 a night for a family bungalow, with breakfast and unlimited wifi included. Meals out cost less than buying groceries at home, and the food is fabulous. Getting laundry done cost about $2 a kilo. I never lifted a finger all the time we were there and it cost less than just living at home. We took boat trips to places like the Emerald Cave, snorkelling trips etc. (You can hire a longtail boat just for your family quite cheaply or join a tour operator's tours.) But actually just being there was an education for kids - everything is so different, you only have to take a walk down the street to see something new and interesting. The people are incredibly kind and friendly.
For a mid-priced option, I'd recommend Port Douglas or a Fiji resort. Port Douglas gives you plenty of options for activities on water or land. Many Fiji resorts have fantastic kids clubs so they're a good option if you have little ones and you want to relax.
For a luxury option, I'd recommend the Cook Islands or French Polynesia. Cook Islands is the most relaxing place in the world IMO - time and cares just melt away. On Rarotonga there are hotels in a range of price brackets. Get to Aitutaki at least for a day trip if you can. French Polynesia is very expensive, so with a family I'd look into an all-inclusive place like Club Med Bora Bora, otherwise your costs will spin out once you arrive.