Interstingly, the word comes from the device used to determine how much to charge for the trip... the tsximeter... one site says It dates back to 1907 and is short for taximeter cabriolet. Taximeter comes from the German word "taxameter," which was a meter used to record distance and fare for horse-drawn carriages. The cabriolet was a light two-wheeled vehicle built in France in the late 1800s. It was drawn by a single horse and had a large leather hood and a leather apron to protect the legs of its one or two passengers from the mud on the ground.
Another site says that taximeters were actually used in ancient Rome. They were devices that dropped metal balls into a container at regular distances and the fare was based on the number of balls accumulated...