Dilute magnesium–aluminum–calcium–manganese alloys have a quick age-hardening response and thus show higher strength after short periods of artificial aging; however, their room temperature formability is rather low. Here, we report that the addition of a small amount of copper (Cu) could substantially increase the Erichsen value of a Mg–0.5Al–0.5Ca–0.3Mn (wt.%) alloy due to the formation of a weakened sheet texture and variation in the distribution of (0002) basal pole figure. Specifically, the 0.03 wt% (300 parts-per-million) Cu-containing alloy showed a large index Erichsen value of 7.4 mm, which was noticeably higher than that of the Cu-free alloy (5.2 mm). Further investigation of the microstructure by transmission electron microscopy revealed that Cu co-segregated with Al and Ca at the grain boundaries, which played a critical role for the formation of the weaker basal texture and finer microstructure in the Cu-containing alloys. Owing to the small amount of the Cu addition, the resultant Mg–0.5Al–0.5Ca–0.3Mn–0.03Cu alloy showed a large thermal conductivity of 131 W/(m·K) in a solution treated condition. Subsequent artificial aging at 170 °C for 8 h (T6) not only improved the thermal conductivity to 141 W/(m·K) but also increased the tensile yield strength from 127 to 181 MPa. The improvement in both thermal conductivity and tensile yield strength by the T6-treatment is associated with the precipitation of densely distributed single-atomic layer Guinier–Preston zones lying parallel to the (0002) basal plane.