It may be derived from the Latin "divinare"; the Oxford English Dictionary
has a quotation in which "divine" refers to "sacred song". Another
possibility is that it derives from the Arabic "diwan", a collection of
poems, based on the fact that many piyutim (poetical prayers) were
incorporated into the liturgy in post-Talmudic times. Still another
possibility is that it comes from the Aramaic "de-avunon" ("from our
fathers"), based on the fact that the three daily prayers were originated
by our forefathers Abraham, Isaac, and Jacob.