We used mailman, but in theory any mailing list software will do.
We created a non-archived moderator approved single subscriber mailing list called our_childs_name@our_domain. Then created a real mailbox called secret_mailbox@our_domain, then setup ACLs in the mailserver to deny anyone but localhost from delivering to secret_mailbox. Then subscribed the secret_mailbox to the mailing list. My wife and I are moderators of the list. When a new email comes in, we get an emailing notifying us theres a message, one click opens the approval page, and one click forwards the mail on, or drops it.
Done and done, all free software and it took only a few minutes with an already working mail server.
If you don't have your own mailserver, then I guess it gets more tricky