I pride myself on being able to find something interesting everywhere in NZ. Here's my list for Invercargill:
the botanic gardens and the museum in the middle where you can see the tuatara, the oldest reptile in the world. That said, it doesn't do a lot (you'll think they are plastic!) but it is there.
walk along the main street and look up, there's some interesting architecture.
Burt Munro's motorbike in the one of the shops on the main street (ask at the visitor centre)
drive down to Bluff for the 'end of SH1 signpost'. It isn't actually the southernmost point of the south island, but stand under it and boggle at the distances to your home. There's also the 'chain' sculpture, the other end is on Stewart island.
if it is the right day you can go up the water tower and see a very long way, as Invercargill is built on a flat plain.
ok, a different scale of interest to some but there's always something. And it is 2 hours to Queenstown, 2 hours to Te Anau and less in the other direction to the Catlins.