Honor Our Fallen Heroes With Dignity Act of 2013 - Directs the Secretary of the Army to permit visitors to leave on any gravesite or government-furnished marker located in section 60 of Arlington National Cemetery any item that: (1) does not present a safety or health concern, (2) is not permanently affixed to such marker, and (3) does not interfere with the normal operations and maintenance procedures of the Cemetery.
Requires the History Office of Arlington National Cemetery: (1) to collect, catalog, and store any other items left on a gravesite, headstone, or marker in section 60; and (2) at the end of each calendar month, to collect, catalog, and store all items left on gravesites, headstones, and government-furnished markers.