The person most directly credited with at least coining the term in relation to the dance was Cab Calloway. A jazz band leader, Calloway had a 1935 recording whose title, "Call of the Jitterbug," helped put the term into popular usage.
The dance style evolved from dance movements in the African-American community. Rather unflatteringly, it was also associated with being drunk and flapping around while intoxicated.
The dance took off in the swing era, made popular not only by Cab Calloway but by other jazz greats such as Benny Goodman. American GIs took the dance with them to England and Europe during World War II, and helped spread it there.
Part of the problem in identifying a specific originator of the dance comes from the fact that many swing-era dances are sometimes referred to as the Jitterbug. In addition to the Lindy Hop, these include the East Coast Swing and Jive dancing.
Regardless of who started it (and most likely it was a group effort), the dance itself is marked by exuberance. Side-by-side dancing, arms waving, legs kicking--it's all in the Jitterbug.