Lift the chair so it is upside down and use a screwdriver to take the seat and backing off the chair. Set all the hardware in a safe spot where it won't get lost.
Remove any staples on the old fabric with a flat head screwdriver. Keep the fabric so you can use it as a pattern for the new material. If designing a new cover when no prior cover existed, measure the seat to see how much fabric you should use and add a few inches to allow for cushioning and stapling.
Pin the old fabric to the new fabric on a flat surface and cut around your pattern to make new pieces for the chair seat and back. Take away the pins after you finish this step. If you don't have a pattern, use your measurements from the prior step.
Get rid of any old cushioning such as batting or foam if you plan to replace it. Once again, use this foam as a pattern for new pieces. If you don't have old cushioning, measure the new foam so that it fits the chair, but is smaller than the cut fabric that surrounds it.
Cut new foam using the old as your pattern. Staple the new foam firmly to the child's rocking chair.
Place the new fabric face down on a table or other flat surface. Set the padded seat on top of it, face down.
Pull the fabric tightly up and around the padded seat and staple it into place on the back of the seat using plenty of staples so the fabric does not fray or peel off. Repeat this process with the cushioning for the back of the chair if necessary.
Put the child's rocking chair back together using the original hardware you set aside earlier.