Your cousin will have to apply for what is called a permanent resident travel document, a one-time PR replacement with which they can return to Canada. Your cousin will have to prove their permanent resident status in Canada and their identity in order to get the document: http://www.cic.gc.ca/english/information/applications/travel.asp
We have information on this as well: http://www.immigroup.com/canadian-permanent-resident-travel-document and of course you can contact us if you wish for paid assistance.