How do I change the birthdate of a child I entered incorrectly? When I follow the directions from Apple Support the box to change the birthday is grayed out

You can't on a child account, from the support page instructions on how to change a date of birth : Update the birth date associated with your Apple ID - Apple Support

Be sure to enter your correct date of birth. After you change it to any age below 13, you can't edit it.

You could try contacting iTunes Support and see if they can/will change it for you : http://www.apple.com/support/itunes/ww or https://www.apple.com/emea/support/itunes/contact.html