I would look into hiking the Routeburn. The national parks outside of Queenstown (Fiordlands & Mount Aspiring) have so much to offer you could literally spend months there without retracing a step, but seeing how you only have seven days I would focus on the highlights.
As David mentioned, the Milford Track is widely regarded as the most spectacular tramp in the area and for good reason. The Clinton Valley is simply stunning and the views from Mackinnon Pass will make the ~$350 you spend on the huts and transportation to and from the hike worth it. That is assuming the weather cooperates and you can still find a permit this close to your departure date. If you're willing to roll the dice on the weather and don't mind the sticker price, go for it. You can visit the DOC's website to check availability, but if you want info on last minute cancellations you're probably better off calling the DOC office in Te Anau directly.
That said, I feel like the Routeburn is your best bet. Rather than burning through four days of your precious time on the Milford you can hike arguably the best high alpine Great Walk and still have plenty of time left over to see Milford Sound and other points of interest in and around Queenstown. And while popular, the Routeburn is more likely to afford you some flexibility in regards to your schedule, allowing you to potentially wait on the weather forecast to pick your dates.
So if I were you I would start by catching a bus or hitching from Queenstown to Milford Sound. If time weren't a factor I would go the hitchhiking route as bus fares can start to add up, but seeing how it is, I would look up fares with Intercity or Tracknet. The drive out to the sound will take 3.5 – 4 hours, but as you're probably already aware, it's worth it. The sound itself is best explored by boat or by plane as there is only a small 2km nature walk in the heart of the sound. So for the budget-conscious like myself, I would suggest doing a kayak tour. The groups are small and the experience is more intimate. Whatever you do, don't skimp here. Getting out in the sound is a must. And if there was one thing I learned about Fiordlands and Milford Sound in particular, it's that it really is more spectacular to visit when it IS raining. So even if the weather doesn't look like it's going to cooperate, don't be discouraged or have it lead you to cancel a trip out on the water.
Considering how long the drive out to the sound can be it's probably worth it to stay overnight before catching a bus back to the Divide to start the Routeburn. When I finished the Milford Track I spent the night at the Milford Sound Lodge, which was a good value for a four-person dorm room. Regardless of whether or not you want to stay overnight you will have to come back the way you came as there is only one road in and out of the sound. To start the Routeburn catch an Intercity or Tracknet bus back to the Divide, which is the western terminus of the Routeburn.
I hiked the Routeburn in both directions: from the Divide to the Routeburn Shelter and from the Routeburn Shelter to the Divide. And while neither approach seemed to have an inherent advantage over the other, by starting at the Divide you set yourself up with an easy return back to Queenstown by way of Glenorchy. Depending on your level of physical fitness the hike can be completed comfortably in two or three days (and some even as a day hike).
The first time I hiked the Routeburn I stayed at both Lake Mackenzie and at Routeburn Falls. Both are impressive by Great Walk standards, but Routeburn Falls is where I am convinced Peter Jackson got his inspiration for Rivendell; the view down into the valley is remarkable. Whether you decide to do it in two or three days just keep in mind that some of the best views are found on the track between Lake Mackenzie and Routeburn Falls, with Conical Saddle (adjacent to Harris Saddle) offering views down into the Hollyford Valley and out to the Tasman Sea.
Past the Routeburn Falls Hut the track drops steadily down to the Routeburn Shelter where you will need to arrange transport to pick you up and take you back to Queenstown for a celebratory Fergburger.
So what's that? Four days? Three if you pushed it. That would still leave you another three days to head up the West Coast to see the glaciers or up the eastern side to Mount Cook assuming the weather gods are feeling gracious. Or hell, just exploring Queenstown and Wanaka. There are so many day hikes in the area that they're just too numerable to name (although Ben Lomond, Rob Roy Glacier, and Roys Peak are considered some of the best).
If the Great Walks don't really appeal to you there a number of more challenging and remote tramps to consider. Some that come to mind include the Rees-Dart Track, the Cascade Saddle Route, the Wilkin-Young Circuit, or the Greenstone Caples. I would highly recommend picking up a copy of Lonely Planet's Tramping in New Zealand by Jim DuFresne, as each of these hikes is profiled and described in detail there. But having hiked all but the Wilkin-Young, I would recommend the Cascade Saddle Route or the Rees-Dart as alternatives to the Routeburn Track. Both are high alpine, and the view from Cascade Saddle is one of the best anywhere on the South Island on a clear day.
So yeah… I don't envy the choices you have to make but as my high school European history teacher would say, "you've got options!" If you've got any other questions feel free to PM and ask away.
Before I sign off, some final wisdom:
1) Take advantage of the hut system. While sometimes mandatory, I found the huts to be part of the tramping experience and something that should not be abandoned in favor of cost savings or trying out your new cuben tent.
2) Sandflies. Some people have skin that react very badly to their bites while others don't really react at all. I was fortunate to find that I was one of the latter, but that didn't mean they weren't a nuisance. Some people swear by picaridin while others told me that they mixed DEET with baby oil (seriously). Regardless, I never found anyone who was safe from the sandflies wrath so I just never bothered.
3) Weather forecasts can be wildly inaccurate. Part of that is due to the region but a lot depends on what weather service you use. Whatever you do stay away from Metservice. I found the most accurate place for forecasts is yr.no. Believe it or not that is a Norwegian weather forecasting service that has relatively reliable forecasts. For those who are better at reading weather patterns than the forecasters themselves there is another site called MetVUW. I have found the radar imaging on that site to be extremely helpful and by the end of my trip was essentially using it exclusively. Bottom line: anticipate rain and expect nothing.
4) There are hundreds of small and remote huts that see little to no use throughout the year. Many require a little digging to discover, but I can steer you to some helpful websites if that sort of thing interests you. A great place to start is the bookstore or a DOC office. Both will stock a large orange coffee table book called "Shelter From the Storm", which provides pictures and historical information about backcountry huts on both islands.
5) DOC is not the same as the NPS. I might catch some flack on here, but I was extremely underwhelmed by some of the DOC staff working at the various visitor centers. While I can't say a bad thing about any of the employees I found on the trail, some visitor center employees lacked even basic knowledge of the tracks in their jurisdiction. Once again my experience varied greatly from location to location, but Te Anau and Wanaka DOC offices will be your best bet for finding knowledgable staff.
6) Beer. A lot of foreigners seem to rave about Tui, but like most beers in New Zealand I found it to be pretty weak. I feel like I'd have a better chance getting a buzz off beer fumes in Oklahoma or Utah than pounding a six pack of Tui, Speights, or Monteith's. The best I found was the Harringtons Rogue Hop.
Kia ora!