To rename a file geodatabase in your project, right-click the geodatabase in the Catalog pane, click Rename, and type a new name. Press Enter when you finish typing. The geodatabase must be part of your project for you to rename it.
Tip:
You cannot rename enterprise geodatabases, but you can rename database connection files (.sde). Right-click the database connection in the Catalog pane, click Rename, type a new name, and press Enter.