Family Telephone Connection Protection Act of 2013 - Amends the Communications Act of 1934 to direct the Federal Communications Commission (FCC) to consider the following types of regulation of inmate telephone service: (1) prescribing a maximum uniform per-minute rate (paid to telephone service providers), (2) prescribing a maximum uniform service connection or other per-call rate, (3) prescribing variable maximum rates depending on factors such as carrier costs or the size of the correctional facility, (4) requiring providers of inmate telephone service to offer both collect calling and debit account services, (5) prohibiting the payment of commissions by such providers to administrators of correctional facilities, and (6) requiring such administrators to allow more than one service provider at a facility so that prisoners have a choice.