Pilgrim women made their coifs out of white linen -- but you can just as easily fashion one by folding and sewing a white handkerchief.
Lay the handkerchief flat on a surface and place the shoelace across the center, so it divides the handkerchief in half horizontally. Fold the handkerchief over the shoelace, and then fold in half again so that you form a square.
Sew up the side of the square opposite the shoelace (this will be the back of the bonnet) and then shape into a coif.
Use the shoelace to tighten it to fit your child's head.