The use of "bus" for both a public transit system vehicle and a set of data transmission wires comes from the Latin word "omnibus," the dative plural of "omnes," or all. "Omnibus" was used as a name for transportation in France, starting in the city of Nantes in 1828. Some sources say that the public system was called a "voiture omnibus" (car for everyone) and others that it comes from a shopkeeper who punned on his own last name, Omnes, so that his store near the terminal of the horse-drawn vehicles was called "Omnes Omnibus" (Everything for Everyone) and the name was attached to the transportation.

In any case, the name "omnibus" stuck when England picked up the idea of the vehicles the next year. If the word had been French rather than Latin, a new name would probably have been given to the vehicles, but as it was, the word "omnibus" acquired a figurative use quite soon; in 1831 Washington Irving could say about the passage of reform legislation, "the Great Reform Omnibus moves slowly." After that, "omnibus" started to mean any kind of collection of dissimilar things.

The first shortening of omnibus to "bus" is recorded in 1832. The words tended to be used interchangeably for several decades, but by 1888 bus was also used as a verb for traveling by bus, while the longer form never became a verb. (In the latter half of the 20th century, of course, "busing" would became a controversial issue in the United States and be made a noun all over again.)

Pioneering electricians called a rod that carried all the power from a source an "omnibus bar," which became "bus bar," and just "bus" as early as 1930. This name carried over to early computers, so that in 1946, the Annals of the Computation Lab of Harvard University could say, "All units in the machine are connected to the central distribution buss over which numbers are transferred from one unit to another with the aid of timed electrical impulses." Despite the variable number of Ss in the word, its meaning remained essentially the same as computer technology expanded.

Sources:
Hibbert, Christopher. The English: A Social History 1066-1945. New York: W.W. Norton & Company, 1987.
http://dictionary.oed.com
http://www.britishbus.co.uk/horsebus.htm
http://www.quido.cz/objevy/autobus.a.htm
http://www.quinion.com/words/articles/omnibus.htm