The same kind of restrictions apply as for copying maps, that is: the relevant database(s) must be open; maps in the source mapset may be open, but must not have unsaved changes.
To copy a mapset:
Select Copy Mapset from the File menu. The Copy Mapset dialog appears:

Fill in the top part of the dialog, specifying the source mapset, database and language, then the bottom part with the target details. (The target mapset need not already exist, but if it does you will be asked to confirm whether you want to overwrite it.) Click OK. The mapset is copied to its new location.