late 14c., bocchen "to repair," later, "repair clumsily, to spoil by unskillful work" (1520s); of unknown origin. Middle English Dictionary says probably the same as bocchen "to swell up or fester; to bulge or project" (though this is only from early 15c. and OED denies a connection) which is from Old North French boche, Old French boce, a common Romanic word of uncertain origin. Related: Botched; botching.
As a noun, "a bungled or ill-finished part," from c. 1600, perhaps from the verb, but compare Middle English bocche "a boil, a pathological swelling, a tumor" (late 14c.), used especially of glandular swellings from the plague, also figuratively "a corrupt person; a rotten condition" (late 14c.), "a hump on a cripple" (early 14c.), which probably is from Old North French boche, Old French boce, a common Romanic word of uncertain origin.