@jmanriquez68 and @zyrq, looks like there's more information here:
http://wordpress.org/support/topic/author-changes-whenever-i-edit-a-post?replies=5
Check the plugin's readme.html for details on this line, which should be added to your theme's functions.php:
remove_filter( 'members_get_capabilities', 'members_remove_old_levels' );
Basically this means that Members has a filter, members_remove_old_levels, which is in place by default, and when the members_get_capabilities filters are run, it steps in and removes the old-style numeric user levels. The above line removes that filter - which means the user levels aren't removed. You can see the difference when you edit a role - when the above line is in place, you'll see level_0, level_1, etc.
What you need to do to get a role in the Author drop-down is give it level_1. Then you need to edit the user profiles with that role. As the post I've linked to above says, it looks like you just have to edit, save, no changes needed. I thought you had to change the role to something else then back again. Anyway, this is really what I thought was a small problem with the plugin. When you edit a role, the changes should be effective immediately without having to go and edit and save every user with that role (which could be quite a task on some installations!). Maybe the plugin needs to refresh the wp_user_level field in the wp_usermeta table when a role is edited?