You have students and you have families. If you delete a student, the family info remains in your LessonMate account (unless you delete that as well). This includes the email address associated with that student.
If a deleted student returns to lessons and you attempt to sign them up with the same info as before, including family details, you're going to get a red message indicating that the email "is taken".
How to sign up a previously enrolled student.
If indeed the student's family is still in your account, then we simply need to attach the student to the existing family. This also applies if the returning student has a sibling currently enrolled in lessons.
click "Add a student"
Enter the student's name
Skip the "Family details" section
Under "Or add to existing family", choose the correct family.
click "add student"