"tall, broad-leafed grass growing in wet places," Old English hreod "reed, rush," from Proto-Germanic *kreut- "reed" (source also of 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.