In this work, using the high-throughput density functional theory calculation method, the influence of 36 alloying elements on the elastic properties of FCC-Cu was systematically studied, and based on the Labusch model, a linear relationship was fitted between the composition and lattice parameters, and the shear modulus. The solid solution strengthening behavior brought about by alloying was studied quantitatively. The results showed that most alloying elements have solid solution potentiality in copper. The change in the elastic modulus and the strengthening effect of alloying on solid solutions were determined by the elements’ positions in the periodic table. In the same period, the alloying elements located in the middle of the period tended to enhance the elastic modulus of copper, while the elements located at both ends have greater solid solution strengthening ability. The predicted results are in good agreement with the experimental values, which provide theoretical guidance for the design of high-performance copper alloys.