1866, "carousal, noisy drinking bout; any merrymaking," represented in England as a typical American English word, perhaps from jam (n.) on pattern of shivaree [Barnhart]. For the second element, Weekley suggests French bourree, a kind of rustic dance. Century Dictionary calls the whole thing "probably arbitrary." Klein thinks the word of Hindu origin (but he credits its introduction into English, mistakenly, to Kipling). Boy Scouts use is from 1920. It is noted earlier as a term in cribbage:
Jamboree signifies the combination of the five highest cards, as, for example, the two Bowers [jacks], Ace, King, and Queen of trumps in one hand, which entitles the holder to count sixteen points. The holder of such a hand, simply announces the fact, as no play is necessary; but should he play the hand as a Jambone, he can count only eight points, whereas he could count sixteen if he played it, or announced it as a Jamboree. ["The American Hoyle," New York, 1864]
Compare jambone "type of hand played by agreement in the card game of euchre."