to dance a jig with someone or cause someone to dance; to make lively or animated.
From 'be-' combined with 'jig,' a type of fast, lively folk dance from Ireland and Scotland, from the verb meaning to dance. Jig itself may derive from Old French gigue.
You almost never see 'bejig' in modern English, but it shows the 'be-' prefix was applied to nearly every verb in Early Modern English. Shakespeare's audience would have understood it immediately, but it sounds completely invented to us now.
Complete word intelligence in one call. Free tier — 50 lookups/day.