This might be useful over the school holidays if you're flexible with dates.
I haven't used this service but it seems to have pretty good reviews: https://www.facebook.com/Transfercar/reviews
From their Facebook page:
Hi New Zealand, we have 100+ FREE rental cars and campervans that are ready to use across the country. Make the most of this opportunity and get on the road, book now! 🚗
Some deals include a free tank of fuel (that doesn't need to be refilled upon return) - look for the green fuel icon.
Most require full drivers licence.
In terms of insurance:
A basic insurance is always included, but the amount of the excess (what you will have to pay if you have an accident) varies depending on the rental operator. This amount is always stated on the vehicle listing page.
You can usually choose to reduce or even cancel this excess for a daily fee that will be at your charge (also called CDW option), which means that in the case of an accident, you will not have to pay anything. This extended cover option can usually cost from $10 to $40 per day depending on the rental operator.
Aucklanders need not apply, I guess.