Turn the stool upside down. Pry up the staples holding the fabric upholstery in place. Once you get them up far enough, grasp them with needle-nosed pliers to pull them out.
Remove the fabric and the old cushion. Lay the old cushion on top of a new piece of foam in your choice of thickness. Trace the old cushion with a felt pen and cut the new cushion out of the foam with a sharp knife.
Set the stool upright. Spray the bare top of the stool with a spray craft adhesive. Press the new foam cushion into place.
Cover the new cushion with the existing fabric or use the old fabric as a template to trace a new piece of fabric. Cut out the new fabric with scissors.
Staple the fabric to the underside of the breakfast stool with a staple gun. Pull the fabric tight on the opposite side from the first staple and staple it. Pull the fabric tight on one side, halfway between the two staples, and staple it. Do the same to the side opposite that staple.
Continue pulling the fabric tight and stapling it in place, alternating sides, until there is one staple every inch. Turn the stool right side up.