The story
Attested since 1922, of unclear origin. There are several suggested origins, but it most likely arose in imitation of the numerous animal-related nonsense phrases popular in the 1920s such as the cat's pyjamas, cat's whiskers, cat's meow, gnat's elbow, monkey's eyebrows etc. A popular folk etymology has the phrase referring to the world champion dancer Bee Jackson. Another suggestion is that the phrase is a corruption of business. The singular bee's knee is attested from the late 18th century meaning something small or insignificant in the phrase big as a bee's knee. Also as weak as a bee's knee is attested in Ireland (1870). It is possible that the bee's knees is a deliberate inversion of this meaning but is not attested. Another possibility is that this is a reference to the (visible) blobs of pollen in bees' corbiculae.
Etymology from Wiktionary, CC BY-SA 4.0.