Honoring Our WWII Merchant Mariners Act of 2013 - Establishes in the Treasury the Merchant Mariner Equity Compensation Fund for payments by the Secretary of Veterans Affairs (VA) of a monthly benefit of $1,000 to each individual who, between December 7, 1941, and December 31, 1946, was a documented member of the U.S. merchant marine (including the Army Transport Service and the Naval Transport Service) serving as a crewmember of a vessel that was operated in U.S. waters by the War Shipping Administration or the Office of Defense Transportation and under contract to, or the property of, the United States. Requires such individuals to: (1) apply for such benefit within one year after the enactment of this Act, and (2) not have received benefits under the Servicemen's Readjustment Act of 1944.

Requires the Secretary to include, in annual budget documents submitted to Congress, detailed information on the Fund's operation.