Child Identity Theft, How does It Effect The Child?

What if there was a misfortune that the mother of a child couldn't use her own social security number on something? What if a family member used a child's social security number for bills, or for rent, but paid all the bills on time? I know identity theft is illegal, but if the child's credit wasn't abused, wouldn't it do good for the child, like build up his/her credit score? When The child comes of age, would he/she qualify for a scholarship?