The correct answer is C) Record of drafts. After reporting a casualty to a mobile offshore drilling unit, the record of drafts must be retained on board. This is because the record of drafts is a critical document that provides information about the vessel's stability and structural integrity, which is important for investigating and understanding the circumstances surrounding a casualty. The other options, such as hull reports, machinery repair book, and anchor record, while important for the overall operation of the vessel, are not directly relevant to the specific requirement for retaining records after a casualty incident.