The design of bone scaffolds for tissue regeneration is a topic of great interest, which involves different issues related to geometry of architectures, mechanical behavior, and biological requirements, whose optimal combination determines the success of an implant. Additive manufacturing (AM) has widened the capability to produce structures with complex geometries, which should potentially satisfy the different requirements. These architectures can be obtained by means of refined methods and have to be assessed in terms of geometrical and mechanical properties. In this paper a triply periodic minimal surface (TPMS), the Schwarz's Primitive surface (P-surface), has been considered as scaffold unit cell and conveniently parameterized in order to investigate the effect of modulation of analytical parameters on the P-cell geometry and on its properties. Several are the cell properties, which can affect the scaffold performance. Due to the important biofunctional role that the surface curvature plays in mechanisms of cellular proliferation and differentiation, in this paper, in addition to properties considering the cell geometry in its whole (such as volume fraction or pore size), new properties were proposed. These properties involve, particularly, the evaluation of local geometrical-differential properties of the P-surface. The results of this P-cell comprehensive characterization are very useful for the design of customized bone scaffolds able to satisfy both biological and mechanical requirements. A numerical structural evaluation, by means of finite element method (FEM), was performed in order to assess the stiffness of solid P-cells as a function of the changes of the analytical parameters of outer surface and the thickness of cell. Finally, the relationship between stiffness and porosity has been analyzed, given the relevance that this property has for bone scaffolds design.