I sort of figured out a way to do this.

1. Add a variant within the Panels module
1. Add a visibility rule so that different variants show at different URLs (clean URLs and paths required), for example, example.com/paneltitle/variant1, example.com/paneltitle/variant1. 
1. Creat a custom module to hook into the menu. There is a great example of a menu module in the module code samples on drupal.org. 

This seems to have worked for me.