|Shakespeare said Caesar had|
"the falling sickness" but that
Othello had "epilepsy."
It's still premature to say where epilepsy itself comes from, but there's a feature on Science Friday's website's "Science Diction" page about the origins of the word. Apparently, "epilepsy" took over describing seizure disorders when the expressions "the falling sickness" and "the sacred disease" fell out of fashion in the early 1600s. Medical historian Howard Markel notes that Shakespeare, in 1599, calls Julius Caesar's seizures symptoms "the falling sickness." Four years later, the Bard's Othello is described as having "fallen into an epilepsy." Those occurrences, Markel argues (on scant evidence, I think), mark the English shift from the old usage to the modern one.
The word "epilepsy" comes from the Greek "epilambanem," which means to "grab," "seize" or "take possession of." Since in those days many people thought epilepsy was caused by spirit possession, the word probably had a double meaning: first, to shake as if grabbed, and second, to be taken over by a spirit.