Mechanical metamaterials have emerged as a new class of functional materials with unusual material properties. An important example of mechanical metamaterials is auxetic material with tunable negative Poisson’s ratio. Such materials are attaching considerable interests due to their promising applications in various engineering sectors. Although enormous studies have been performed to present negative Poisson’s ratio properties in various materials, research work on the auxetic behavior of graphene reinforced metal nanocomposites is still at its early stage. The present work develops a graphene origami structure to tune the negative Poisson’s ratio in graphene reinforced copper nanocomposites. Extensive molecular dynamics simulations are conducted to study the tuning mechanism of negative Poisson’s ratio of graphene/Cu composites. The adaptive intermolecular reactive bond order (AIREBO) potential, embedded atom method (EAM) potential, and Lennard-Jones (LJ) potential are employed to simulate the C-C covalent interaction of graphene, the interactions between Cu atoms, and the van der Waal (vdW) interactions between graphene and Cu matrix, respectively. Various graphene origami structures, including Miura pattern, Waterbomb pattern, and Yoshimura pattern, are formed with the assistance of surface hydrogen functionalization in predesigned areas of the graphene sheet. The shape of the graphene origami can be tuned and controlled by changing the content and width of H adatoms in the creases. Afterwards, the designed graphene origami is embedded into a Cu matrix to achieve the tunability of negative Poisson’s ratio. Our results demonstrate that a higher content of graphene origami in composites can lead to a larger negative Poisson’s ratio. In addition, the ambient conditions have considerable effects on the auxetic behaviors of graphene/Cu metamaterials. A greater negative Poisson’s ratio of composites can be achieved in higher temperature and pressure environments. It is also found that bigger folds in graphene origami can help the metal metamaterial to reach more efficient auxetic properties.