Make a 2-1/2"-diameter ball of polyester fiberfill. Tightly wrap with perle cotton or string to shape into a flattened ball.
Tear fabric scraps into 1/2"- to 3/4"-wide strips that are 12-20" long. Tack one end of a torn strip at center bottom of flattened ball. Tightly wrap strip around ball; secure end with two or three small whipstitches. Add strips one at a time until ball is completely covered. Wrap contrasting torn strips in an X, tacking ends of each at center bottom. Sew button at center of X on top.