Buckling-restrained braces (BRBs) have excellent hysteretic behavior while buckling-restrained braced frames (BRBFs) are susceptible to residual lateral deformations. To address this drawback, a novel self-centering (SC) BRB with Basalt fiber reinforced polymer (BFRP) composite tendons is presented in this work. The configuration and mechanics of proposed BFRP-SC-BRBs are first discussed. Then an 1840-mm-long BFRP-SC-BRB specimen is fabricated and tested to verify its hysteric and self-centering performance. The tested specimen has an expected flag-shaped hysteresis character, showing a distinct self-centering tendency. During the test, the residual deformation of the specimen is only about 0.6 mm. The gap between anchorage plates and welding ends of bracing tubes performs as expected with the maximum opening value 6 mm when brace is in compression. The OpenSEES software is employed to conduct numerical analysis. Experiment results are used to validate the modeling methodology. Then the proposed numerical model is used to evaluate the influence of initial prestress, tendon diameter and core plate thickness on the performance of BFRP-SC-BRBs. Results show that both the increase of initial prestress and tendon diameters can obviously improve the self-centering effect of BFRP-SC-BRBs. With the increase of core plate thickness, the energy dissipation is improved while the residual deformation is generated when the core plate strength exceeds initial prestress force.