"tall, broad-leafed grass growing in wet places," Old English hreod "reed, rush," from Proto-Germanic *kreut- "reed" (cf. Old Saxon hraid, Old Frisian hriad, Middle Dutch ried, Dutch riet, Old High German hriot, German Ried), with no known cognates beyond Germanic.
Meaning "musical pipe made from a reed stem" is from late 14c. (reed-pipe is from c.1300). As part of the mouthpiece of a musical instrument it is attested from 1520s. Meaning "a reed instrument" is from 1838.