Responsible Homeowner Refinancing Act of 2013 - Requires the Federal National Mortgage Association (Fannie Mae) and Federal Home Loan Mortgage Corporation (Freddie Mac) (government sponsored enterprises or GSEs), in carrying out the Home Affordable Refinance Program, to adopt specified criteria pertaining to: (1) borrower eligibility, (2) representations and warranties, (3) prohibition on up-front fees, (4) alternative streamlined methods to determine the value of a property, (5) the purchase or guarantee of any new mortgage resulting from the refinancing of an eligible mortgage, and (6) guarantee fees.

Requires the GSEs to notify all borrowers with a mortgage owned or guaranteed by a GSE about the Program and its eligibility criteria, and inform borrowers of the website required below.

Directs the Director of the Federal Housing Finance Agency (FHFA) to establish a single website where borrowers may: (1) determine their potential eligibility for participation in the Program, (2) see a complete list of and links to qualified lenders, (3) use a mortgage refinance calculator to calculate potential payment savings based on different interest rates, and (4) obtain tips on refinancing their loan.

Directs the Director of FHFA to issue guidance to require the GSEs to make their refinancing guidelines consistent to ease the compliance requirements of qualified lenders, and in particular with respect to loans with less than 80% loan-to-value ratio and closing cost policies of the GSEs, which regulations or guidance shall be put into effect not later than 90 days after the enactment of this Acts.