In this work, we study gravity duals of conformal field theories with boundaries-known as AdS/BCFT correspondence, put forward by Takayanagi Phys. Rev. Lett. 107, 101602 (2011)-within a scalar-tensor theory, as proposed by Horndeski in 4D Int. J. Theor. Phys. 10, 363 (1974). We consider the case of 3D gravity dual to 2D BCFT, take a Gibbons-Hawking surface term modified by Horndeski's theory, and find the corresponding 3D (Bañados-Teitelboim-Zanelli) black hole solutions. We analyze the effects of Horndeski gravity on the profile of the extra boundary for the black hole by using an approximate analytic solution. Performing a holographic renormalization, we calculate the free energy and obtain the total entropy and corresponding area, as well as the boundary entropy for the black hole. In particular, the boundary entropy found here can be seen as an extension of the one proposed by Takayanagi. From the free energy, we perform a systematic study of the 3D black hole thermodynamics and present, among our results, an indication of the restoration of conformal symmetry for high temperatures. Finally, we present a study on the influence of Horndeski gravity on the Hawking-Page phase transition where we can see the stable and unstable phases throughout the plane of free energy versus temperature.