It's going to be cold and probably wet!
Inverness Castle - not that old as it was rebuilt after the Jacobites blew up the original; Inverness Museum and Art Gallery; Ness Islands (walk); Old High Church; Eastgate Shopping Centre (!!); get out of Inverness and head to the Culloden Battlefields, Clava Cairns, Loch Ness, Cairngorms (Aviemore), Ben Nevis although this isnt that close.
There are some nice walks near Inverness but it might be too cold or take a trip in the Morray Firth and look for bottlenose dolphins.
This is a good website for whats on each week : http://www.inverness-scotland.com/whats-on-inverness.asp about 6 years ago
From my experience, Inverness isn't really that great. The surrounding regions, however, are packed with things. Loch Ness, of course. Invermoriston is beautiful - go for a walk around the river and down to the folly.
At the other end of the Loch is Fort Augustus which is also beautiful.
There are some lovely Glens and mountains in the region too which are stunning when snowcapped in the Winter, especially to the east and south. Ben Nevis has been mentioned, also Glen Coe. Cullodon is probably worth the trip just because of the intensity of the site. I guess it depends on what kind of things you are looking to do. almost 6 years ago