Thank you for bringing up this very important question, Jane. You have already read the excellent input offered by others. Here I offer an additional two bits.
Checking about the attitude was mentioned and I agree. I think it is pretty important to pay attention to the child’s attitude. In a case like this the attitude in coming is usually fairly evident – did he come in order to get off the hook, a manipulative move? Is there true repentance? When true repentance is evident, like Peter said, the child is usually ready and even eager to pay the consequences. Then too, we need to check our own attitudes when meting out the consequences. One of “you had this coming” is certainly not redemptive!
But going a bit deeper yet, I feel it is imperative that we listen carefully to discern what drove the cheating or lying. What lay behind it? What created the ‘need’ to cheat/lie? I deviate now to a scenario where the child got caught in the act rather than coming to confess, but I think the same principles apply: I’m thinking of a child who struggled desperately in his studies, was caught cheating, and promptly delivered a sound spanking. Perhaps that was the correct response, but I cringed inwardly, feeling sure the child was driven by a desperate fear of failure. I suspect there could have been more redemptive responses. While I would never endorse winking at such an offense, without looking at root causes and giving the child a reason to believe that you are out to do all you can to help him succeed (and not just academically), the child may well only end up resolving to be more careful to not get caught next time. Often the surface sin is not actually the issue needing most to be addressed.
I think of a student of mine who I noted was beginning to struggle with respect, especially in one class. When I asked her in private, “What is going on in your heart?” she took completely me by surprise by sobbing out the truth, “I am so afraid.” “Afraid of what?” “Afraid of failing (this class and) the grade.” She was not even close to failing, but this conversation allowed me to respond redemptively to what really lay behind the unacceptable surface actions. If we look only at the surface evidence without considering what lies behind it, our success will often be limited.
Another comment: each child is unique. Much as we would wish to, we cannot live by a set of rules that says “always do it this way”. What will be redemptive for one child may well only harden another’s heart. Although that approach could feel like a wishy-washy approach, and it requires me to invest a lot more, the call is to rely on the Holy Spirit, who leads us into truth. He alone really knows the heart of each child. As such, relying on Him is also key.