Taxi comes from Tasso (not from Tax like some people think). Tasso (know even as Taxius) was an Italian Family who maid Mail service big in XIII. The Tasso family woud provide mail service in Italy, France, Germany and Spain from XIII to XVIII. They start transporting people around XIV.
Hackney comes from an old French word for Horse (haquenée). That meant a calm walking horse, particularly suitable for ladies. This got initially translated into English to simply mean a fine quality horse, in the 1300s. Carriages for hire would advertise their hackey horse, and soon the word became synonymous with a carriage for hire before the year 1400.
Cab has French origins, too. It comes from cabriolet, which was a type of small horse-drawn carriage. Cabriolet dates from about 1763. Shortened to 'cab' and used as a generic term for vehicles for hire the phrase appears to date from 1826.
Taxi came in with the first motorised vehicles for hire (German and French), in the late 1890s. It comes from the taximeter that keeps track of how much you need to pay for the journey. The 'taxi' itself comes from a Greek word, taxideyo, which means 'to travel'.
