Honor Those Who Served Act of 2013 - Revises veterans' burial benefits provisions to allow a person to request a headstone or marker from the Department of Veterans Affairs (VA) to commemorate an eligible individual if such person is: (1) the decedent's next of kin; (2) a person authorized by the next of kin to make such request; (3) a personal representative authorized by the decedent to make such request; (4) in the case of a decedent for whom there is no person under (1) through (3) identified, a state veterans agency, a military researcher, a local historian, a genealogist, or other person familiar with methods necessary to prove the identity of the decedent; or (5) in the case of a veteran who served on active duty at least 62 years before the date on which the headstone or marker is requested, any person.

Allows for the request of a headstone or marker without an emblem of religious belief.