To ensure the safety of any minor travelling on a cruise ship, some cruise lines may ask any parent taking the minor on holiday, to be able to show they are the parent, in cases where the family surname on the passports may be different.
They may ask the adult to show documentation proving they are the parent such as birth/marriage certificates or divorce documents. These need to act as evidence of the name changes and that the individuals are indeed parent and child.
Each cruise line may have different rules around these situations, so we advise you to check with the cruise line's website at the earliest opportunity, so you have time to find the necessary documents.
An example of what a cruise line may ask you to provide (Royal Caribbean, Celebrity and Azamara Cruises policy) can be found here.