Boost

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early 19th century (originally US, in boost (sense 2 of the verb)): of unknown origin.

wiktionary

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Of unknown origin. The verb is first recorded 1815; the noun, 1825. Compare Scots boost(“to move; drive off; shoo away”), bost, boast(“to threaten; scold”), Middle English boosten, bosten(“to threaten”).

etymonline

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boost (v.)

"to lift or raise by pushing from behind," 1815, literal and figurative, American English, of unknown origin. Related: Boosted; boosting. As a noun, "a lift, a shove up, an upward push," by 1825.