Ok, I'm awake now :)
By preference of my proudly Kiwi DH we travel Air New Zealand. They are really fabulous, but often a bit more expensive.
Last time we travelled out here (3 years ago), we flew Dublin-London-LA-Auckland.
Pros: I'm pretty sure it's the shortest time - the stop off is just for crew change and refueling
Cons: It's LA. We were travelling premium economy, so we got a little bit of extra help (no need to queue for immigration - they let you off first with a little card, so you can go back and do immigration once all the economy class passengers are finished). Otherwise, it's off the plane, queue for ages, sit in holding room (with free tea/coffee/snack etc) and get back on the plane. Even if you travel business class (which we did once years ago) and you want to use the lounge you have to go through immigration, customs, through the baggage claim, physically out of the airport building, back in, through passport control etc before getting to the lounge.
Alternatives: Air NZ through Hong Kong. Much nicer process, you get to wander around the duty free shops etc. Altogether a much more pleasant experience (thinking back, I now have no idea why we didn't chose that way when we flew over this time, there must have been some reason).
Emirates: Depending on where you are flying from this may be an option. My parents fly Dublin - Dubai - Brisbane - Auckland. Emirates are usually the cheapest option for them at least.
We usually try to get the whole thing over in one go. While it's pretty horrendous to do 30 hours of travelling all in one go, then at least it's over. The thoughts of going out of the airport, checking in somewhere, unpacking, only to have to repack and do another very long flight a few days later, just doesn't appeal to me with young children.
Sorry for the epic post, I'll probably be back with more when it occurs to me later!
Is it for a move or a holiday? (I think that might inform my decision a bit on airlines/stopovers etc too)