Oddly enough for AKL June is usually fairly sunny although it's also cold (maybe 8-14 celsius).

If you want to stay in Auckland there's a couple of things to do depending on what you're into.

1) Go up the Sky Tower if it's sunny - great views of the city.
2) The Auckland Museum - Great if you're into history and cultural, also a good place to go if the weather isn't quite so nice.
3) If you're feeling athletic take a Ferry to Rangitoto Island and climb to the summit.
4) Try the restaurants! Compared to most international prices, Auckland's food is fairly cheap and always fresh especially the seafood.

You can also do day trips out of Auckland (either with a car or a tour company). Rotorua and Waitomo Caves are quite popular.

Hope this helps