Buy 2-inch-wide cotton or nylon webbing at a crafts store. Measure the length and width of the chair seat, and add 2 inches. Cut enough strips of each size to weave your seat.
Lay strips across seat from front to back, leaving 1 inch on both sides; fold ends under 1/2 inch, and secure with a staple gun; use a rubber mallet to cover staples with upholstery nail heads as you go.
Place strips over these from side to side; staple and finish one of the sides. Weave the loose ends of the strips across the seat. One at a time, tug strips until taut and fasten.