For the first time, an alternative way of improving the stability of Cu-based thermoelectric materials is proposed, with the investigation of two different copper chalcogenide–copper tetrahedrite composites, rich in sulfur and selenium anions, respectively. Based on the preliminary DFT results, which indicate the instability of Sb-doped copper chalcogenide, the Cu1.97S–Cu12Sb4S13 and Cu2−xSe–Cu3SbSe3 composites are obtained using melt-solidification techniques, with the tetrahedrite phase concentration varying from 1 to 10 wt.%. Room temperature structural analysis (XRD, SEM) indicates the two-phase structure of the materials, with ternary phase precipitates embed within the copper chalcogenide matrix. The proposed solution allows for successful blocking of excessive Cu migration, with stable electrical conductivity and Seebeck coefficient values over subsequent thermal cycles. The materials exhibit a p-type, semimetallic character with high stability, represented by a near-constant power factor (PF)—temperature dependences between individual cycles. Finally, the thermoelectric figure-of-merit ZT parameter reaches about 0.26 (623 K) for the Cu1.97S–Cu12Sb4S13 system, in which case increasing content of tetrahedrite is a beneficial effect, and about 0.44 (623 K) for the Cu2−xSe–Cu3SbSe3 system, where increasing the content of Cu3SbSe3 negatively influences the thermoelectric performance.