Queenstown is a must-see place in New Zealand. I would argue that it's the most beautiful place in New Zealand - closest to the rivalling great places to see in the whole south island.
The best time to rent is in Summer. Although of course people push to visit Queenstown in winter because of it's magnificent views and fantastic winter sports and ski fields.
Summer is when the rates are down and there's less traffic. Aside from less traffic, there's an 80% chance you'll get parking in the city. The winter is beautiful but somewhat unbearable for people from warmer climates.
The best places to rent would be around the city center. The other places like Frankton, Sunshine bay and Queenstown East claim they are 5 km from the city but in reality, you'd need a car to get around. If you don't, you'd have to pay $4 per way with the Queenstown bus. It is unreasonably priced especially if you only need it once or twice. If you are a single traveller, you can buy the weekly bus ticket... which is still priced slightly higher than any other nice city in the world.
Kelvin heights is a nice, up-market suburb. It has a barely used golf course and you can use the free wifi in their cafe for a couple of hours. Unfortunately, Kelvin heights would seem like a world away from the city - it's 10 km from the city and you really need a car.
Better rent a place that is furnished. This way, you don't have to bring a bunch of stuff and take them all with you when you leave.
Queenstown New Zealand