Moderating the quiz lets you allow extra attempts for individual students as well as multiple students at once. This option also allows you to grant extra attempts for students who have not yet taken the quiz. For timed quizzes, you can grant extra time for an attempt. If your quiz options allow students to view results only once after each attempt, you can also let students view quiz results one more time.
If you need to provide an extension for a student to complete the quiz outside of your originally programmed access/due dates, you can assign the quiz to the specific student, while leaving the original access/due dates for the rest of the course.