Bi0.5Na0.5TiO3 (BNT)-based lead-free materials are important for piezoelectric actuator, and several researchers have studied the effect of B-site complex ion doping on strain in (Bi0.5Na0.5)TiO3–SrTiO3. In this work, a paraelectric perovskite Sr(Zn1/3Nb2/3)O3 (SZN) with B-site complex structure was introduced into 0.80(Bi0.5Na0.5)TiO3–0.20SrTiO3 (BNTST) to investigate the phase structure and electrical properties as well as the field-induced strain behavior. The results showed that SZN substitution decreases the rhombohedrality 90-γ and induces the transition from dominant ferroelectric to nonergodic relaxor by shifting its TF-R to lower temperatures. Moreover, the field-induced ferroelectric domains cannot remain stable at room temperature when SZN substitution is large than 1.0 mol%. These behaviors induced the transition between nonergodic relaxor and ergodic relaxor, which contributed to its large strain and related properties. In this work, this material gave the largest bipolar strain of 0.43% and large normalized unipolar strain of 505 pm/V at the SZN content of 2 mol% under 8 kV/mm, and showed good temperature stability up to 100 °C. The above encouraging results may be helpful for further investigation of BNTST-based ternary systems in search of a potential Pb-free piezoelectric material.