New Zealand South Island in winter

How about going over Christmas break? That would be their summer and much better for hiking. The major hikes like the Milford and Routebourne Tracks are not really open for hiking in their winter---bridges are down, and avalanche is possible. The hikes in Mt Cook Aoraki National Park would be under snow. At the north tip of the island, Abel Tasman NP might be OK, but the beaches might not be conducive to swimming, or sea kayaking.

We went in January two years ago and it was wonderful. We hiked in three of the places mentioned above, and loved it. The Milford Track in particular is unique and spectacular. Mt. Cook Aoraki offers scenery like the Alps, with peaks and glaciers, and a fun hike with 2200 well-built steps. At the north end we stayed at a remote lodge reached by water taxi--it looks a lot like Hawaii there. Google up Lochmara Lodge for photos. We hiked on the Queen Charlotte Track and kayaked in the bay. It felt like paradise. Lake Tekapo, with its gorgeous sapphire-blue water, offers more trails for hiking or mountain biking. Then there is the Otago rail trail, and the peninsula by the same name with penguins.

I don't know how much of this would be enjoyable in their winter. That is ski season in the mountains.

We found the South Island very easy to navigate, with a wealth of reasonable accommodations. Mostly we stayed in motels, which offer self-catering facilities.

While it's true that the national, multi-day hikes like the Routebourn, Milford, etc. will be closed during the winter months, I think you and your family would still enjoy a visit to the South Island in July/August.
Abel Tasman National Park is located at the top of the South Island (tropical climate), and has enough beautiful hiking trails to keep you busy for a month. They run a water taxi service from the little village of Marahau which will deposit you at any number of trail heads from where you can do a one-way hike back to your starting point.
The Otago peninsula near the university town of Dunedin has some great walks too, and the scenery is universally terrific.
I'd also put in a plug for the Akaroa peninsula about an hour east of Christchurch. The French themed town is a delight, and the surrounding bays and inlets offer many opportunities to just meander for a day - some nice hikes there too, in fact we'll be heading back there ourselves in February.
Queenstown is the jewel of the South Island. Even though it will be in the midst of the ski season in the surrounding mountains the town itself will very much be open for business. It's the adventure capital of NZ. If your family likes adrenaline rushes there are endless opportunities to try to kill yourselves in a variety of creative ways. It's located in a beautiful setting next to Lake Wakatipu - lots of Lord of the Rings film locations, and a side trip up the lake to Glenorchy is an enjoyable excursion too. You can at least access the start of the Routebourn track from there. As I recall the first part of the track is pretty level - you don't start to ascend into the mountains until 4 or 5 miles along. There's a good chance that it will be snow free in August.
You should still be able to drive the road from Te Anau to Milford Sound, but do check the weather & road conditions carefully before venturing out - that particular stretch is one of the most dangerous in NZ, due mainly to inattentive drivers transfixed by the scenery and not paying enough attention to the narrow and winding two lane road.
Bottom line is that I wouldn't hesitate to schedule a trip during their winter months.

You may find temps below freezing overnight and struggling to reach double figures (Celsius) during the day for July/August in NZ South Island. Hiking is out. Skiing is in. You may also be faced with transport disruptions.

My husband and I spent almost a week in Queenstown in July, 2 years ago, and we absolutely loved the time of year we chose. I think it depends on what type of weather you are comfortable with. We love the snow, so it was perfect for us. It was cold, but nothing quality winter clothes couldn't take care of. You can go skiing, although we didn't find the skiing to be near the same as what is offered in somewhere like Colorado. We also did some snowshoeing, which was wonderful. There are so many winter activities going on in Queenstown in July. You wouldn't be lacking in things to do. One of our best memories was taking a flight to Milford Sound, taking a cruise, and then flying back to Queenstown. I can't describe how beautiful it was to fly so closely over the snow covered mountains!! My vote is to go now when you can, even if that means in July :)

Yes, Queenstown will be quite active. Two south-hemisphere winter trips and never saw snow below 3000 feet elevation. Golf was even in the picture for some, but skiing, paragliding, jet boating and many other adrenaline activities on offer. Especially enjoyed a boat ride on the 100 year old steam boat TSS Earnslaw to a farm with dinner and a show of the working Border Collies herding sheep under whistle command. If skiing is in the picture, look for a day on the Tasman Glacier with a ski plane. Otherwise, several ski areas a short bus ride from town.

The South Island stretches almost 600 miles from north to south, so temperatures will vary greatly depending upon where you are. The area around Abel Tasman NP will have very mild weather in August. Queenstown's average daily high will be in the 50's - especially the later in August you go. The Canterbury Plain (around Christchurch) tends to be drier than the west thanks to the effects of the southern Alps, which knock down most of the bad weather before it progresses further east. Point being that there are plenty of hiking opportunities (and outdoor activities in general) during the winter months in NZ.

Thank you all so much! I really appreciate the thoughts and tips. I am eager to start planning this trip with all of these great recommendations.