"move air," Old English blawan "blow, breathe, make an air current; kindle; inflate; sound a wind instrument" (class VII strong verb; past tense bleow, past participle blawen), from Proto-Germanic *blæ-anan (cf. Old High German blaen, German blähen), from PIE *bhle- "to swell, blow up" (cf. Latin flare "to blow"), possibly identical with *bhel- (2) "to blow, swell" (see bole).
Meaning "to squander" (of money) is from 1874. Sense of "depart suddenly" is from 1902. Slang "do fellatio on" sense is from 1933, as blow (someone) off, originally among prostitutes (cf. blow job). This usage probably is not connected to the colloquial imprecation (1781, associated with sailors, e.g. Popeye's "well, blow me down!"), which has past participle blowed. Meaning "to spend (money) foolishly and all at once" is 1890s; that of "bungle an opportunity" is from 1943. To blow over "pass" is from 1610s, originally of storms. To blow (someone's) mind was in use by 1967; there is a song title "Blow Your Mind" released in a 1965 Mirawood recording by a group called The Gas Company.