by Randy Carey:
State law issue.
by John Burnett:
Safe deposit boxes are rented, not owned. Some states allow the renter or lessee to designate an individual who can, under that state's laws, access the box after the lessee's death (or death of the last surviving lessee in the case of co-lessees) to remove contents (state law may limit what can be removed) and, in some cases, terminate the lease and close the box.
There are such wide differences in state laws governing safe deposit boxes that you really have to research those laws and any potential regulations.