The influence of Cu and Hf concentrations on structure, martensitic transformation, mechanical and functional behaviour was studied in Ti49,2Hf1Ni49,8-xCux and Ti40,7Hf9,5Ni49,8-xCux (x = 1, 5, or 10 at. %) alloys. The temperatures and hysteresis of the martensitic transformation increased on a rise in Hf content and decreased on an increase in Cu concentration. An increase in Hf concentration suppressed the B19 phase formation, while a rise in Cu content promoted the B19 phase formation. An increase in Hf and Cu concentration led to the solid solution hardening that increased the strength and the yield limit for dislocation slip. The strain up failure increased on a rise in Cu concentrations. The variation in Hf and Cu concentrations did not affect the recoverable strain, which value was close to 5.3 ± 0.5% on cooling and heating under 200 MPa, while the irreversible strain decreased on a rise in Hf and Cu content.