Without re-hashing what's already been suggested (at least, not too much), here's what comes to mind:
- A simulated medical emergency: Improvising a splint for a "broken" leg, or even Ace bandaging a simulated sprain. Bandaging. How to use a tourniquet, and when not to.
- A communications exercise with the NZ equivalent of FRS radios. Dirt cheap, 1/2 Watt, UHF, 1/4-mile range, license-free, it's all line-of-sight propagation. Looks like NZ has something similar:
wikipedia link. Maybe setting up a relay man to provide comms over a ridge between two valleys, something like that. That would involve teamwork, at least.
- Canoeing can be a pretty fun team activity. A lot depends on the local waters, how the teams are arranged, etc.
- Geocaching (with GPS, rather than map & compass as you initially wrote) is always a crowd-pleaser.
I dunno after that. After watching LOTR too damn many times, it's hard for me to get images of Middle Earth out of my mind when I think of rural NZ. Maybe a Ringwraith SERE scenario would be taking things too far.