As per
this thread my no.1 co-driver and i did exactly this a few years back.
The whole of the South Island is, pretty much, a driving paradise. Immediately outside of Christchurch if you head into the Port Hills there are good driving roads no matter which direction you go. Get a local map and just head towards any winding blacktop - there's plenty of it within 10-15 minutes drive of ChCh.
Further afield head south towards Queenstown and drive the Crown Range, it runs between Lake Wanaka and Arrowtown/Queenstown. Stupendous piece of road that - sadly, on that trip, we were not in an MX5. But have every intention of returning for that reason in the future
The drive out to Milford Sound is also very memorable.
West from Ch Ch you can also find Arthurs Pass which is another legendary piece of road. Of course, the condition of these passes is weather dependant, but in the warmer month like Feb you shouldn't have any issue.