A letter template must have a Data source of Assignments or People to be used for assignment reminders. If Data source is People, then the template must have a sub-letter with a basis of Assignments. For SMS messages the Preferred sending method must be SMS/text message, and for e-mails it must be Any mail or E-mail. If these conditions aren't met, VSys won't show the template as an option here.
Your assignment reminder templates should probably include something about the assignment itself.