There are two circumstances where a birth can be re-registered:

• Adding the father's details if not recorded at the initial registration - if the parents are unmarried and would like the natural father's details added to the birth record, they need to re-register the birth so that a new birth record can be created to replace the original one
• If the child's parents have married since the child was born - if the natural parents have married each other since the birth, they need to re-register the birth so that a new birth record can be created showing the child as a child of the parents' marriage

For more information see GOV.UK - Adding a father's name to a birth certificate.