What permissions does a role need for the user to be assigned as the author of a post?

You probably need to add 3 capabilities:

• edit_published_posts
• publish_posts
• delete_published_posts

but at the very least, you need to add publish_posts as that is the capability WordPress uses to determine what users are shown in Authors drop-down.

Note: Changing capabilities is stored in the database so the recommendation is to modify these values via a plugin and only when the plugin is activated or deactivated rather than on every request.