Australia, New Zealand & the South Pacific - Need advice on traveling with a 3 year old to New Zealand

We hope to visit New Zealand from Dec 19, 2009 to Jan 9, 2010. First thing we saw was that the flights had 3 stops and involved travel time of almost 24 hours. We need to keep our costs down, around $5000 for the three of us, and we are wondering if there is the possibility of getting almost 24 hours out of the plane during the trip (preferably where there is a beach) so we are not so jet lagged on arrival. Some flights went through the Cook islands or French Polynesia. Can anyone suggest possibilities for a decent trip from DTW to AKL or some nearby alternate airports?

Air NZ has fares that allow routing via south pacific islands and "free" stopover enroute. So for example you'd fly to LAX with UA or US, then LAX to APW (Samoa) or TBU (Tonga) or RAR (Cook Islands), stop, then resume to AKL (Auckland).

Air NZ also flies from Honolulu - you'd need to fly UA to HNL.

PPT (Tahiti) can also be done, but I expect will be more expensive on Air NZ now they no longer fly LAX-PPT themselves. Otherwise check out Air Tahiti Nui fares - they fly LAX-PPT and PPT to AKL, or Air Pacific (stop at NAN Fiji).