"stupid person," by 1951, American English, from dumb (adj.). The Disney musical cartoon is from 1941; in it the elephant's name is a mocking nickname based on Jumbo.
Entries linking to dumbo
dumb adj.
Old English dumb, of persons, "mute, silent, refraining from speaking or unable to speak," from Proto-Germanic *dumbaz "dumb, dull," which is perhaps from PIE *dheubh- "confusion, stupefaction, dizziness," from root *dheu- (1) "dust, mist, vapor, smoke," also expressing related notions of "defective perception or wits." The -b has probably been silent since 13c. Related: Dumbly; dumber; dumbest. Of animals, "lacking in speech," hence "without intellect" (c. 1200).
The fork in meaning probably comes via the notion of "not responding through ignorance or incomprehension." The Old English, Old Saxon (dumb), Gothic (dumbs), and Old Norse (dumbr) forms of the word meant only "mute, speechless;" in Old High German (thumb) it meant both this and "stupid," and in Modern German this latter became the only sense (the sense of "mute, speechless" being expressed by stumm). Meaning "foolish, ignorant" was occasional in Middle English, but the modern use in this sense (since 1823) seems to be from influence of German dumm, especially in Pennsylvania German.
dumb-cake ..., n. A cake made in silence on St Mark's Eve, with numerous ceremonies, by maids, to discover their future husbands. [Century Dictionary]
Applied to silent contrivances, hence dumb-waiter. Dumb ox "stupid man" is by 1756; dumb-bunny "stupid person" is college slang from 1922; dumb blonde "woman seen as incapable of comprehending anything complicated" is by 1936.
jumbo adj.
"very large, unusually large for its type," 1882, a reference to Jumbo, name of the London Zoo's huge elephant (acquired from France, said to have been captured as a baby in Abyssinia in 1861), sold February 1882 to U.S. circus showman P.T. Barnum amid great excitement in America and great outcry in England, both fanned by Barnum.
"I tell you conscientiously that no idea of the immensity of the animal can be formed. It is a fact that he is simply beyond comparison. The largest elephants I ever saw are mere dwarfs by the side of Jumbo." [P.T. Barnum, interview, "Philadelphia Press," April 22, 1882]
The name is perhaps from slang jumbo "clumsy, unwieldy fellow" (1823), which itself is possibly from a word for "elephant" in a West African language (compare Kongo nzamba). OED suggests it is possibly the second element in mumbo jumbo. Century Dictionary says "The name was given as having an African semblance." As a product size, by 1886 (cigars). Jumbo jet attested by 1964. Jumbo was accidentally killed near St. Thomas, Ontario, Sept. 15, 1885, struck by a freight train while the circus was loading up to travel.