van der Waals (vdW) atom-surface potentials can be excellent benchmarks for atomic structure calculations. This is especially true if measurements are made with two different types of atoms interacting with the same surface sample. Here we show theoretically how ratios of vdW potential strengths (e.g., C(3)(K)/C(3)(Na)) depend sensitively on the properties of each atom, yet these ratios are relatively insensitive to properties of the surface. We discuss how C(3) ratios depend on atomic core electrons by using a two-oscillator model to represent the contribution from atomic valence electrons and core electrons separately. We explain why certain pairs of atoms are preferable to study for future experimental tests of atomic structure calculations. A well chosen pair of atoms (e.g., K and Na) will have a C(3) ratio that is insensitive to the permittivity of the surface, whereas a poorly chosen pair (e.g., K and He) will have a ratio of C(3) values that depends more strongly on the permittivity of the surface.