I'm sorry to hear you are having to deal with this extra hassle and possible displacement of residence when you just lost your mother. The terms of the lease will govern and sometimes there is in fact a provision that discusses successors in interest for both parties, though that usually applies to obligations not rights. But in general, the only person who has legal right to stay in the apartment is the actual tenant listed and any listed occupants (even if not officially liable on the lease).
Answered on Jun 05th, 2013 at 7:54 PM