A number of reserves and viewing areas combined into one exceptional park, Rakiura National Park covers much of Stewart Island, the third largest (behind the two main ones) in New Zealand. It encompasses about 85% of the island, excluding the main community of Oban (where you'll most likely be looking to find accommodation if you're staying for a while).
The reasons for staying are plenty. The Rakiura Track may be your most complete way of exploring what the island has to offer, with a 32 km (20 mile) length that takes a good three days to circuit; there are huts and campsites spread out along the track that need to be booked ahead of time with the Department of Conservation. But you don't have to just follow the track, for there are plenty of hiking trails otherwise that will bring you into contact with kiwi, penguin, and plenty of other endemic creatures. The Southern Circuit and the Northwest Circuit are even lengthier tramping options. No matter your hiking choice, you'll no doubt encounter anything from sand dunes to wetlands to rocky inlets and pristine coves.
For much more on accommodations, route details, and preparation information, check out the official Department of Conservation page.