I visited South Island in March 2002 for 2 weeks, which was towards the end of their summer. The weather was lovely, sunny and warm though New Zealanders that we met on the journey told us it had been pretty wet up till then. We had some rain only on three of the days and then only a couple of hours each time. The climate in the South Island is similar to the UK so I would say anytime between the end of October and March would be the best.
The roads have mostly good surfaces and apart from a lot of hills and some interesting bridges that are for both trains and cars not too different to driving in the UK.The best thing about driving in NZ apart from around Christchurch is the absolute lack of traffic, miles and miles of driving without seeing another car. There was snow on Mount Cook but it is an alpine area and the roads were clear.

We were there at the end of August, and then most of September which I suppose is still classed as winter moving on to spring. Weather was very good most of the time although pretty cold as you can imagine. Aside from the west coast of the South Island were it rained consistently for a day, we didn't really experience any bad weather apart from gales down near Te Anau. This seriously affected any journeys to Milford Sound, and meant we didn't get to see the Sound on this trip. Allow yourself plenty of time down there if you want to see Milford, just in case the weather is bad.

The good thing about going at this time is that it is far quieter and prices are reasonable.

Driving is a dream. As the other poster has indicated, you do go for miles without seeing another car.

On our RTW trip New Zealand was my favourite country, and a close 2nd for my Wife to Nambia.

We went to the South Island at the end of January last year and had lovely sunny, warm weather most of the time, like a really pleasant British Summer. The only time the weather was poor was when we travelled up the west coast to Franz Josef. It poured with rain and was very misty, but not cold but I believe that it's quite often like that on that coast.
The roads were great to travel on all of the time. Those narrow bridges with a railway track down the middle were a bit scary though,

Agree with what has been said so far - driving was better than expected with empty good roads. We were there in October and had changeable weather, some gales, some sunshine and drizzle on the west coast - they don't call it rain forest for nothing. You are supposed to carry snow chains on the Milford Road in October - you can hire them at a local garage quite cheaply - but though there was still snow at the top of the pass, the road was clear and we didn't need to use them. We did come across some gravel roads round the Catlins and Otago Penisula, though they are well maintained and drivable in a normal car, as long as you don't go too fast. They can be avoided and do make sure your car hire allows you to use them, if you do decide to follow them. We loved the South Island, the scenery and wildlife was fantastic.