In my continuing search for things that drop on New Year’s Eve, I have found:
A possum (Brasstown, NC)
A lighted beach ball (Carolina Beach, NC)
An orange (Orange County, CA)
A new Ford Edge (Buffalo, NY)
A carp (St. Feriole Island, WI) 2008 Update: New article
Bologna (Lebanon, PA) 2008 Update: New Article

and a long list of things from a giant red shoe carrying a drag queen (Key West, FL) to a wrench (Mechanicsburg, PA) on Wikipedia.

New York:
The Ball Drop in Times Square stems from an even older ritual in England when a descending Time Ball was a popular method of signaling noon to allow pocket watches and nearby at sea ships chronometer’s to be reset daily. This tradition continues onward today in Washington D.C. at the United States Naval Observatory where it still drops a Time Ball down a flag pole every day at noon.

The latest ball measures 6 feet in diameter. The lighted Ball descends 77 feet in sixty seconds. The Ball is a geodesic sphere, six feet in diameter, weighing approximately 1,070 pounds. The Ball is covered with 504 Waterford crystal triangles, 696 multicolored Philips light bulbs, 96 high intensity strobe lights, and 90 rotating pyramid mirrors.

Atlanta: Giant Peach
New Orleans: Gumbo Pot
Mt. Olive, NC: Giant Pickle
Port Clinton, Ohio: 600-pound fiberglass walleye

Let me know if you know of anything else.