In recent years, the design and development of new high-performance alloys based on first principles have received extensive attention. However, there are few reports on the structural design and thermodynamic properties of Cu-Zr alloys at nanoscale. In this work, based on the crystal structure characteristics of CuZr₂, 12 kinds of Cr-doped CuZr₂ structures are designed and optimized by the method of Cr atom doping through the first-principle calculation based on the density functional theory, and 6 kinds of mechanically and dynamically stable doped structure models are found. By calculating the electronic structure, elastic properties and hardness of the CuZr₂ and its dynamically stable Cr-doped structure, it is found that the studied objects have all energy bands that pass through the Fermi energy level and are metallic. The main contributors to the metallic properties of the CuZr₂ are the p and d orbital electrons of Zr, while the main contributors to the metallic properties of the 6 dynamically stable Cr-doped CuZr₂ structures are the p and d orbital electrons of Cr and Zr. Meanwhile, CuZr₂ has symmetrically distributed spin electrons, which do not show magnetism externally. However, the doping of Cr atoms increases the elemental species of the matrix. In addition to the difference of spin electrons brought by the d-orbital electrons of Cr atoms, the doped Cr atoms destroy the symmetrical distribution of electrons with different spin directions in the p- and d-orbitals of Zr atoms in the matrix, so that the designed 6 dynamically stable Cr-doped CuZr₂ structures exhibit ferromagnetic properties with magnetic moments ranging from 0.303 to 5.243μ_B. In addition, it is found that Cr atoms can improve the mechanical properties of CuZr₂. When the Cr atom is used to replace the Zr atom in the matrix, the elastic modulus and hardness of the material can be improved, and when the Cr atom is used to replace the Cu atom in the matrix, the machining properties of the material can be improved due to the reduction of hardness. The datasets presented in this work, including the band structure, density of states, and phonon dispersion frequency, are available from https://www.doi.org/10.57760/sciencedb.j00213.00122.