It indicates that the database is open but not visible as an available file in Open remote. Usually it would be a data source file that is not openned directly by the user but is referenced from another file.
Thank you for your post.
Ron Cates is spot on with his assessment. The half circle indicates that the preference to not show in the open remote dialogue is checked. I am including a screenshot of the window showing the preference in FileMaker for additional clarification.