Family Telephone Connection Protection Act of 2017 This bill amends the Communications Act of 1934 to direct the Federal Communications Commission to adopt or continue in force regulations on the use of inmate telephone and video services that enable persons incarcerated in correctional institutions to originate calls at payphones, telephones, or video kiosks designated for the personal use of inmates. The regulations must: (1) prescribe variable maximum per-minute compensation rates depending on factors such as carrier costs and the size of the correctional facility, (2) prohibit per-call or per-connection charges, (3) require providers of inmate telephone and video service to offer both collect calling and prepaid debit account services, (4) prohibit or limit the payment of commissions by such providers to administrators of correctional facilities based upon the revenue of the service, (5) require such administrators to allow more than one service provider at a facility so that inmates have a choice, and (6) prohibit or substantially limit any ancillary fees that are in addition to the per-minute rate and connection charge.