Usually a slang term baffles those who search for its source.
By the time it has become popular, the person who invented it and the occasion leading to the invention are forgotten and cannot be located.
Jitney acquired popularity throughout the United States in 1914 and 1915 as a slang term for a five-cent piece, a nickel. The meaning was then transferred to the passenger vehicle, other than a streetcar, for which the fare was, originally, five cents.
A correspondent of the late Dr. F. H. Vizetelly, quoted by him in his A Desk-Book of Idioms and Idiomatic Phrases, suggested that jitney might be a corruption of jetnee in a catch song of the French-speaking Louisiana Negro:
Mettons jetnee dans li trou
Et parcourons sur la rue
Mettons jetne.e-si non vous
Vous promenez a pied nu!
Which is freely translated thus:
Put a jitney in the slot
And over the street you ride
Put a jitney-for if not
You’ll foot it on your hide.
The explanation is plausible, and jetnee itself may be a corruption of the French jeton, a token, counter.