Ummm Queenstown....my home town!
As Mat says the drive between Christchurch and Queenstown is fantastic. The usual places to stay along the way are either Tekapo, or 'The Hermitage' (at the base of Mt Cook).
As for rental cars, the best for new cars are;
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www.avis.co.nz
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www.hertz.co.nz
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www.budget.co.nz
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www.europcar.co.nz
If you are after something cheaper (usually Jap imports etc), then try the following;
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www.rentadent.co.nz
If you fly into Auckland airport there are literally dozens of cheap rental car outfits within striking distance of the airport that mainly target the budget traveller and backpackers. Rent A Dent (above) is the only name I can think of though. Some Lonely Planet guides can now be downloaded in PDF format, so perhaps you could check the Lonely PLanet website and see if the NZ guide is one of them (for example, the China LP can be downloaded in PDF format for $2 per chapter).
Likewise, Christchurch has a good selection of cheap reatals as well, but I can't think of their names. If you fly into Queenstown then try;
www.queenstownrentalcars.co.nz
The post above says rental cars are a bit ropey in NZ, but as always you just get what you pay for..... if you want a good quality near new car, then stick to Hertz, Avis, and Europcar.
Being summer, Christmas time is of course the time that New Zealanders take holidays and head off to the beach or lake for a few weeks. So, as a couple of people have mentioned already, you really do need to think about booking accomodation
now. Try;
www.jasons.com/new-zealand
As for the weather.... if you can delay your holiday until February, you will find the weather much better!