The small Hawaiian instrument, whose name literally means jumping flea, and whose flea may have been a man: Edward Purvis, the small, lively British officer in King Kalakaua's court nicknamed Ukulele.
Documented
The Purvis account
The story favored by Merriam-Webster's etymologists: uku, flea, plus lele, jumping. Edward Purvis, a petite and sprightly British former army officer in Kalakaua's court, bore the nickname Ukulele, played and promoted the machete, and the nickname transferred to the instrument. Hawaii had the word before it had the instrument: an 1865 dictionary defines ukulele as a cat flea.
Assessment: Documented: the 1865 dictionary entry and the Purvis account. The word outdates the instrument in the language, which settles the order of events.
Stories
The queen's version
Queen Liliuokalani's alternative: ukulele from uku, gift or reward, and lele, to come, the gift that came here. Linguists note the two uku are distinct Hawaiian words, and the gift reading is the lovelier and less probable story, preserved because a queen told it.
Assessment: Stories: the royal etymology is documented as Liliuokalani's. Probability is documented as elsewhere.
Interesting claims
The renamed colony
The pattern: three Madeiran luthiers, Nunes, Dias, Santo, built the instrument; Hawaii renamed it, branded it, and exported it back to the world as Hawaiian culture, so successfully that Madeira is a footnote in its own invention. The 1915 San Francisco exhibition craze completed the laundering.
Assessment: Interesting: a colonial-era import became the icon of the colonized nation's identity. The name did the work.
Sources and further reading
Merriam-Webster etymology, ukulele: uku, flea, plus lele, jumping; the Edward Purvis nickname account (lexicographers' favored story)
Hawaiian dictionary of 1865 defining ukulele as cat flea (the word predating the instrument)
Queen Liliuokalani's alternative reading, the gift that came here (documented royal etymology)
The Madeiran machete arrival, 1879; luthiers Nunes, Dias, Santo (the documented instrument history)
Worth asking
Is the jumping flea the player's fingers, or the man the instrument was named for?