mid-14c., prosperen, "be successful, thrive, advance in any good thing," from Old French prosperer (14c.) and directly from Latin prosperare "cause to succeed, render happy," from prosperus "favorable, fortunate, prosperous" (source also of Spanish and Italian prospero).
This is perhaps etymologically "agreeable to one's wishes," traditionally regarded as from Old Latin pro spere "according to expectation, according to one's hope," from pro "for" (see pro-) + ablative of spes "hope" (from PIE root *speh- "prosperity" (see speed (n.)). Or, if the compound is older, from Proto-Italic *pro-sparo-, from PIE *pro-speh- "to thrive," with second element from PIE *sph-ro- "thriving" (source also of Old English spōwan "to prosper;" again, see speed (n.)). The rarer transitive sense of "make to prosper, promote the success of" is from 1520s.