For your alloted time, I would focus on only one island, either north or south and then probably focus on one area, particularly if your time is more towards the 3 day mark.
Both have a lot of options. Prepare for rain wherever you go in NZ.
Also, keep in mind many tracks can only be tramped in one direction. The Routeburn is an exception here.
South Island
The South Island has more alpine terrain and more remote areas particularly in fjordland. For anything in the South Island, try and fly direct into Christchurch. You will lose a few hours by going into and connecting via either Auckland or Wellington.
You could either do a longer track or break it up experiencing a few different areas of the south island with shorter hikes
For example, spend a couple days around Mount Cook area and do some dayhiking, a day or two in Kaikoura where you can
do some hiking or take a marine wildlife trip where you can see albatross, petrels, whales, etc. Possibly also see Arthur Pass NP enroute.
You could stay the whole time in Queenstown and do day excursions including the Routeburn (albeit in a LONG day).
or split it up tramping a few days and then in town a few days.
The Milford Track would be great (assuming you can get hut reservations) but keep in mind you need to commit some time here, I think 5 days, plus getting to an from the area, so the time needed here may preclude you from doing it.
The Dart and Heaphy Tracks are also worth looking into.
North Island
The North Island has some wonderful terrain, especially around Tongariro. Check volcano status here, but assuming you can walk along the track, you can do the entire circuit there plus the two volcanos (Ngauruhoe and Ruapehu) in a few days. Highly recommended. Prepare for snow on Ruapehu.
You can also visit a place called Rotorua, a place known for its geothermal activity and Maori cultural center.
Mount Taranaki is also a cool peak to think about climbing, but is more remote and doesn't have a lot of wild terrain about it like Tongariro does and is surrounded by more farmland. It also rains more there and you may not even get a chance to see the peak, but would be worthwhile to investigate!
Taranaki (aka Mt Egmont) was featured in the Last Samurai (filling in for Fuji) since that film was filmed in NZ. Stunning mountain when visible.
