To save the active map in a different location on the database:
Select Save Copy As from the File menu, or (from 4.622) press Ctrl+A. The Save Map As dialog appears:

Fill in the dialog, specifying the mapset,language and map number, then click OK to save the map. If this operation would overwrite an existing map, you are asked to confirm that that is what you want to do.
Using the "Save Copy As" option does not change the map being edited into the new map, you are still editing the original map, and will still be prompted to save your changes to that map. If you do not want to change your original map then, after using the Save Copy as dialog, close the current map edit window without saving your changes and open the newly saved map if desired.