"to make utterly confused, put into muse or reverie, muddle, stupefy," from be- + muse (compare amuse); attested from 1735 but probably older, as Pope (1705) punned on it as "devoted utterly to the Muses."
word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause, seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by;" see by (prep.)). The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).
The Old English prefix also was used to make transitive verbs and as a privative prefix (as in behead). The sense "on all sides, all about" naturally grew to include intensive uses (as in bespatter "spatter about," therefore "spatter very much," besprinkle, etc.). Be- also can be causative, or have just about any sense required. The prefix was productive 16c.-17c. in forming useful words, many of which have not survived, such as bethwack "to thrash soundly" (1550s) and betongue "to assail in speech, to scold" (1630s).
late 15c., "to divert the attention, beguile, delude," from Old French amuser "fool, tease, hoax, entrap; make fun of," literally "cause to muse" (as a distraction), from a "at, to" (from Latin ad, but here probably a causal prefix) + muser "ponder, stare fixedly" (see muse (v.)).
Original English senses obsolete; meaning "divert from serious business, tickle the fancy of" is recorded from 1630s, but through 18c. the primary meaning was "deceive, cheat" by first occupying the attention. "The word was not in reg. use bef. 1600, and was not used by Shakespere" [OED]. Bemuse retains more of the original meaning. Greek amousos meant "without Muses," hence "uneducated."