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

This operation does not make a new session folder — the existing session folder and relevant contents are renamed. If your 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.