How do I determine which version of a royal person’s name to use in my works-cited-list entry?

Use the version of the name given in your source. For example, if the source gives the author’s name as Sarah, Duchess of York, then use that form and list the entry under “Sarah.” But if the source gives the author’s name as Sarah Margaret Ferguson, use that form and list the entry under “Ferguson, Sarah Margaret.” 

If you are using several sources, and the name is treated variously, choose one form—using a reference source to determine the preferred form, if possible—and consolidate the entries under that name, following the guidelines in section 2.1.1 of the MLA Handbook. You may add an alternative form in parentheses after the name:

Ferguson, Sarah Margaret (Duchess of York).