This trip appears to be booked in heavily discounted economy. "N" fares are among the lowest buckets around and earn nothing or very little in programs other than the "home" program of the operating airline.
As you state that you don't fly regularly and there might even be years without flight activity at all, primary concern should be making sure any points do not expire. So what you need is a program with a generous expiration policy. That rules out all programs with a fixed expiration policy and leaves AA, BA and QF who let points expire based on activity - AA and QF require activity once every 18 months, BA once every 36 months.
Next, the program of choice should let you earn points on NZ domestic flights as you state that's what you fly most often, and - if possible - a way to earn points with non-flying activity to keep the balance from expiring and contribute to your total earnings.
My advice would be to go with QFF - even though there's far more attractive programs out there in oneworld, it's the only program delivering on the last two requirements.
First, you can
earn QF points on domestic Jetstar flights. QFF is the only OW program that lets you earn on JQ.
QFF is also the only OW program that offers a
co-branded points-earning rewards credit card in New Zealand. Co-branded credit cards are the best source of points for non-frequent flyers, plus by using it you prevent your points from expiring even if not flying for extended periods of time.