Use Session > Rename to give the session a new name. A dialog will appear asking for the new name.

This operation does not make a new session folder—the existing session folder and relevant contents are renamed. If the session was not saved before a rename operation, it will be saved automatically and then renaming will continue.

Ardour's Session > Save As operation will not make a new copy of the session folder and its contents. All it does is create a new session file.