The rule is short: the NZeTA is attached electronically to the passport used in the request, and you must travel on that passport. Everything else follows from that.
Choosing which passport
- It must be a passport from a visa-waiver country or territory — otherwise the NZeTA is not available on that document
- If only one of your passports is eligible, that is the one to use
- If both are eligible, use the one you will actually be carrying and checking in with
Where it goes wrong
Dual nationals are refused boarding for this more often than for any problem with the authorisation itself. The airline checks the passport in your hand against the record. If the NZeTA is attached to your other passport, the system finds nothing, and nothing is what the check-in agent acts on.
If you renew one of them
A new passport means a new NZeTA, even though the authorisation would otherwise last 2 years. This applies to whichever passport the NZeTA is attached to.
New Zealand citizens
If one of your nationalities is New Zealand, you travel on your New Zealand passport and do not need an NZeTA at all. Requesting one on your other passport does not change that.
Source: Immigration New Zealand — NZeTA
More on the NZeTA
Children and infants
Yes. Every traveller needs an NZeTA in their own name, including infants who are travelling on a parent's lap and do not have their own seat.
Australians
Australian citizens need neither a visa nor an NZeTA for New Zealand. Australian permanent residents are different: they must hold an NZeTA before boarding.
Work and study
Arriving under the visa waiver lets you study for up to 3 months. It does not let you work. What counts as work is wider than most travellers expect.
Medical treatment
If the purpose of the trip is medical treatment, Immigration New Zealand requires a Medical Treatment Visitor Visa. The NZeTA does not cover it.
NZeTA or visa?
The NZeTA covers short visits from visa-waiver passports. Longer stays, work, study beyond 3 months and medical treatment need a visa instead.
Which one do I need?
Similar names, very different rules. Which scheme applies to which destination, what each one costs, and the mistakes that come from mixing them up.