Servicemember Housing Protection Act of 2013 - Amends the Servicemembers Civil Relief Act to protect a surviving spouse who is the successor in interest to a servicemember who dies from a service-connected cause while in military service against a mortgage foreclosure or residential lease termination. Extends such protection for the one-month period following the death of the servicemember.

Allows the termination of an existing lease by the lessee, without penalties, when the lessee is assigned to or otherwise relocates to federal quarters or military housing. Requires, in such case, the lessee to deliver to the lessor written notice of such termination and a letter from the servicemember's commanding officer indicating such relocation.