Hi Ian,
Winter is definitely the preferred time to visit Chennai. Regarding hotels, I think the Taj Coromandel, Chola Sheraton and Park Sheraton are the 3 "usual" high end hotels. There's also hotel Trident and Le Meridien IIRC, which are near the airport and far from the city center. Cheaper ones might include the GRT Grand Days (considerably better than its counterparts in the US) and Quality Inn Aruna.
You should easily be able to fit in a trip to Mahabalipuram, where you'd find the last remaining one of the seven Shore Temples (built by the Pallava Kings?). The beach there is quite nice too. Chennai's own beach is touted as the second longest in the world after Miami.
In Chennai, the Santhome Bascillica and St. Thomas Mount where the hill's namesake was originally buried might also be of interest to you.
There are many temples in the city which are very old but still in use.
If you want to get a taste of the local music scene, try the Music Academy or Narada Gana Sabha (expect Indian vocal artists or classical instrumental performances).
Good places to get handicrafts and local art would be Kalpa Druma in Nungambakkam and VTI on Mount Road near Spencer Plaza. VTI is vaguely government authorized to sell products made from sandalwood (fragrant wood famous in Mysore and south India). Their prices are reasonable and you don't have to (or can't) haggle.
Definitely try some south indian food. Saravana Bhavan (which has a couple of branches in London too) is a good example. Don't miss Dosas (crepes) and Sambhar (lentil soup, usually mixed in with rice).
Other places you consider visitng include the hill stations of Ooty and Kodai (you'd need to take a short flight / long train ride and then transfer by car).
Bangalore and Mysore aren't far either. Mysore is famous for it's palace and resident Maharaja.
If you're into wildlife, consider visiting Mudumalai National Park and the Thekkady area. You should see plenty of elephant and a tiger or two if you're lucky. Making this trip would definitely take up at least 3 days though.
Good luck and feel free to ask more questions.