change child's name

Try logging in via a computer and accessing your account settings to try and change your childs name on the account.

You will need to set up your child with their own Microsoft/Xbox account. It will save tears at a later date, when they find out they can't move their saved game data.

Follow the steps in the link posted by RedCarGamer

So try this way, not by Xbox but accessing
Microsoft Family by browser. Just log in with your account, Select Manage my child's profile info. For the child’s account that you want to change, select Edit this child's personal info and then follow the instructions. See if now you will be able to change
his name.

Usually in Xbox you cannot change your name, even you with the main account, this stuff needs to be changed directly in the Manage Microsoft Account through the website.

No, we have a family account. I am the primary person and he is under me.

Did you create a separated account for him? (this means a different microsoft account/email address)