late 14c., coyte (Anglo-French), "a flat stone thrown in a game," later also a ring of iron used the same way (15c.); a word of uncertain origin; probably from Old French coite "flat stone," which is perhaps literally "cushion," and a variant of coilte (see quilt (n.)).
Quoits were among the games prohibited by Edward III and Richard II to encourage archery. In reference to the tossing game played with iron rings, from mid-15c.
Formerly in the country, the rustics not having the round perforated quoits to play with, used horse-shoes, and in many places the quoit itself, to this day, is called a shoe. [Joseph Strutt, "Sports and Pastimes of the People of England," 1801]