According to the Podictionary: In August of 1826 Stanislas Baudry in Nantes, France introduced the idea of public transport to the world. He did it a little bit by accident actually because what he was really trying to do was make it easier for customers to find their way to the public baths he owned and ran as his business.
The idea of giving people a ride was a big success but things didn’t exactly go as planned. The trouble was that everyone kept getting on and off the bus at all the stops between downtown and his bath house. Stanislas was no dope though; he gave up on the public baths and instead began charging a flat fare to ride as far as a passenger wanted to go along the route.
The idea was so popular that people in other places heard about it and within 6 years busses were set up in London, Paris, Bordeaux, and Lyons. Except they weren’t called busses, instead they were called "vehicle for all" which in French was voiture omnibus. In Latin omnis means "all."
According to The Oxford English Dictionary Stanislas Baudry chose the name voiture omnibus because he knew of a local tradesman who was named monsieur Omnès and used the word play Omnès omnibus to publicize his business. You could imagine a plumber named Everett putting up a sign advertising "Everett for everyone." Word travels fast—faster than public transit anyway—and it only took 3 years for the word omnibus to arrive in London. That’s half the time it took the omnibus service to arrive there.
I guess once omnibuses began operating in London, the people riding them could a afford to be a little lazier than they’d been before and so they quickly—in the same year, 1832 according to the OED—began abbreviating omnibus down to bus