Go into the folder where the test or quiz is located. Click on the arrow beside the test and then click on "Edit the test options".
You can leave the due date, but you must uncheck “Do not allow students to start…” for the student to be able to take the test/quiz.
Then you must set Availability dates for it to disappear from students. This will help to ensure no one else is able to take the test.
Then give the selected student(s) access after the availability dates by clicking "Add User or Group"(see picture below).
This will give only that student access to the test and other students will not see it.