The little plastic tube on the end of a shoelace is called an aglet. It stiffens the lace, prevents its woven fibers from fraying, and makes the end narrow and firm enough to pass through a shoe’s eyelets.
Modern aglets are usually made from heat-sealed plastic, though shoelace tips can also be metal. The word has older roots: “aglet” developed from Old French terms related to a needle or small pointed object. Earlier aglets were often made from metal, glass, or stone and could be decorative as well as practical.
Aglets are not limited to shoes. Similar sheaths can finish the ends of drawstrings, cords, and ribbons. A related term, “aiglet” or “aiguillette,” is more often associated with decorative cord tips, especially in historical clothing and military dress.
The wrong options are playful inventions rather than alternate names. “Eyelet” refers to the hole or ring in the shoe through which the lace passes; the aglet is the stiffened tip that helps the lace get there.