I will be staying at a hotel off of Michigan Avenue over New Years, and would appreciate any suggestions as to inexpensive parking.

what is "inexpensive"? This is important as the Chicago downtown area is VERY expensive parkingwise.

Where off Michigan Ave? How far are you willing to walk?

The 900 N. Michigan Garage (behind the shopping center with Bloomingdale's in it) has discounted parking when you spend a certain minimum. Enter on Walton Street. It's the best short-term deal I've found in that immediate area. Parking is really at a preimium these days - I'll bet the areas just west and east of Michigan Avenue have lost half of their surface lots in the last 5 years.
If you are parking the car and leaving it, try negotiating with the lot for a weekend price with no in and out privileges. Many lots won't be attended over the holidays, so if you are hemmed in, you are stuck, but this may be a way to get your costs down.
In the downtown, the best price is at the city-owned Monroe Street garage (enter on Columbus, north of Monroe); about $12 a day and open all the time. Be sure to pay as you leave or re-enter at the things that look like ATMs in the lobby.

You may want to reconsider going far for your car when your visit is finished. The weather outlook for your stay seems to be very, very cold. Taking patg's advise about the underground parking might play out with the advantage of your car being kept out of the wind and in a warmer temperature. You may have to take a cab to get your car if winds and weather stay this cold, regardless. It might be better to pay the hotel's fees. There is nothing in the inexpensive parking category in downtown Chicago, by my definitions of inexpensive. I have also seen high rise parking where wind is so bad that it will take quite a while to get your car unfrozen.

We just returned 12/24/01 from downtown Chicago stayed at Embassy Suites on North State, parkind with in out priveleges was 34.00. I parked across the street with no in or out priveleges for 26.00. It was very cold and it was worth saving 9.00 parking across the street it would not have been worth it if the parking garage was any farther!! Cab fares are expensive so park at the hotel or across the street.