Sulfated zirconia (SO42-/ZrO2) has been reported as an efficient and stable solid superacid catalyst for normalizing i-butane to n-butane, and may potentially substitute Pt-based catalysts. In this work, sulfated zirconia with Cu addition (Cu-SO42-/ZrO2) is prepared through a one-step method and exhibits comparable performance to commercial Pt-Cl/Al2O3, with an i-butane average conversion of 43.8% and n-butane average selectivity as high as 86.1%, while maintaining a stable n-butane yield for up to 120 hours. The relationship between Cu-promoter-induced physicochemical properties and catalytic performance for sulfated zirconia catalysts is investigated in detail, with the help of XPS, NH3-TPD, pyridine-IR, and density functional theory calculation. A volcano-type relationship is found between Cu content and n-butane yield, where n-butane yield first increases from 32.23% to 37.71%, and then decreases to 29.56%, with the gradual increase of Cu content. The moderate Cu species additive is beneficial to release Zr Lewis acid site from sulfate and increase its Lewis acid density, which accounts for its excellent catalytic selectivity for i-butane normalization. The excess Cu promoter covers the exposed Zr Lewis active sites, leading to the decayed i-butane conversion. Therefore, Lewis active sites density is regarded as descriptor to corelate n-butane yield, accounting for the volcano relationship between Cu content and n-butane yield for Cu-SO42-/ZrO2. This work demonstrates the feasibility of Cu modification and provides insight into the promotion mechanism, which may be extended to other solid superacid catalysts beyond sulfated zirconia.