Change children's Apple ID
http://appleid.apple.com/ is the correct place.

It's been awhile since I did this, but as I recall you will need to delete it as the secondary email then you can edit the primary.

Also, it's easiest if you turn off FindMyiPhone on the device before starting this process.  After all changes have been made and you are logged back into iCloud on the device using the new appleid then be sure and turn FindMyiPhone back on. (FindMyiPhone is a security feature and one it's purposes is to make it difficult for someone without the appleId details to log out. Changing your appleId can confuse the secuirty features, solution is to turn it off during this process.)

Also realize that once you have made the change at appleid.apple.com, the are multiple places in settings on the phone to logout and log back in under the new: iCloud, iTunes and App Store, facetime, messages, Game Center, and likely email needs setup changes.

Edit (based on OP's comment below): Apple states that "If your email address ends with @icloud.com, @me.com, or @mac.com, it's already an Apple ID." Therefor it's not going to be possible to make this appleid have a different primary email address.  If you want an appleid using your son non @icloud.com address you will have to create it.  In order for that to happen you may have to delete it as the secondary on this account.  There is logic in no edit button appearing for the primary. If you made this appleid differnt, how could the @icloud address still be an appleid?