There's no doubt that outdoor enthusiasts visiting New Zealand will want to spend most of their time on the South Island. The mountains are bigger, the landscape is more dramatic, and there don't seem to be any people around to bother you. But don't count out the North Island!
I spent nearly a week at Te Urewera National Park during the winter. I can certainly see the advantages of visiting during the summer—warm weather, swimming, boating—but for someone who just wants to get out and enjoy the outdoors, winter is the time to visit! During the winter you will feel like you have the park to yourself and the others you run into will be similarly interested in having a great outdoor weekend without the tourist nonsense.
Miles of hiking trails will take you to waterfalls, viewpoints, lakes, swimming holes, and the unique flora of New Zealand. There are plenty of day hikes and also a loop around Lake Waikaremoana that you can spend days on (spend nights in the cabins built along the way). You'll need to bring your own supplies, but you'll have a roof and a bed platform at least.
I recommend finding a place that overlooks the entire lake and sitting there on a windy day. You simply can't explain how amazing it is to watch the wind whip up the water into ghostly swirls and walls of water that seem to just spring out of the lake.