There are a few different options for synonyms for the term "Cayman Islander." First and foremost, you could simply use the term "Caymanian," which refers to someone from the Cayman Islands specifically. Depending on the context you're using the term in, you may also be able to use a more general term like "Caribbean native" or "Caribbean islander." Other possible synonyms could include "West Indian" (though this term could refer to people from other Caribbean islands as well), "Antillean" (referring specifically to the Antilles region which includes the Cayman Islands), or "Islander" (which is a more general term often used to refer to people from various islands around the world).