Safer Neighborhoods Gun Buyback Act of 2013 - Authorizes the Director of the Bureau of Justice Assistance to make two-year grants to states and local governments for subgrants to gun dealers, or to gun dealers directly, to conduct gun buyback programs under which gun dealers shall be issued smart prepaid cards to purchase specified guns (listed in this Act) from individuals wishing to dispose of them.

Prohibits: (1) such an individual from using such a card to buy a gun or ammunition, and (2) a merchant from accepting such a card to sell a gun or ammunition.

Requires a state or local government to: (1) use 10% of grant funds to recycle the guns received from dealers to make street signs, energy efficient washing machines, car parts, energy efficient refrigerators, or other steel parts such as railroad or metro tracks; and (2) use not more than 10% for administrative costs of the program.

Requires a gun dealer participating in the program to: (1) pay an individual 125% of a gun's market value, as determined by the Director; and (2) deliver guns received to the closest Bureau of Alcohol, Tobacco, Firearms and Explosives [ATF] office (for grantees) or to the state or local government (for subgrantees) within 60 days of receipt. Allows grant funds to be used to provide incentives to gun dealers to participate.