Middle English sight, from Old English sihð, gesiht, gesihð "thing seen; power or faculty of sight; aspect; vision; apparition," from Proto-Germanic *sekh(w)- (source also of Danish sigte, Swedish sigt, Middle Dutch sicht, Dutch zicht, Old High German siht, German Sicht, Gesicht), stem that also yielded Old English seon (see see (v.)), with noun suffix -th (2), later -t (14c.).
The meaning "perception or apprehension by means of the eyes" is from early 13c. The meaning "device on a firearm to assist in aiming" is from 1580s. A "show" of something, hence, colloquially, "a great many; a lot," (late 14c.). As "something that calls forth glances of shock, amusement, etc., a shocking spectacle," by 1862.
Sight for sore eyes "welcome visitor" is attested from 1738; sight unseen (adv.) "without previous inspection" is from 1892. Sight gag is attested by 1944. To feel or know something at first sight is from c. 1300. From the firearm aiming sense come in (one's) sights; have (one's) sights set on something. To keep out of sight is from late 14c.; to be out of (someone's) sight is from c. 1400.