If I haven't already done so, welcome Mcgriff. If your mother plans on going on any more cruises, I would highly advise her to go ahead and apply for a passport. Don't wait until you book the next cruise. Just go ahead and have it ready to go. Since your cruise originates and terminates within the U.S., a CERTIFIED copy of a birth certificate and a driver's license will suffice (for now). This is directly from RCCL's website regarding travel documentation:
U.S. citizens on closed-loop cruises (cruises that begin and end at the same port in the U.S.) will be able to enter or depart the country with proof of citizenship, such as a birth certificate and laminated government issued picture ID, denoting photo, name and date of birth. A U.S. citizen under the age of 16 will be able to present either an original or a copy of his or her birth certificate, a Consular Report of Birth Abroad issues by DOS, or Certificate of Naturalization issued by U.S. Citizenship and Immigration Services.