The electrical state of the interface between a kaolinite-dominated clay sample and aqueous electrolyte solutions was characterized using low-frequency conductance measurements. From these measurements, the ζ-potential and surface conductivity contributions from the diffuse and non-diffuse parts of the electrical double layer were obtained. The suspensions were studied as a function of volume fraction, electrolyte concentration, and electrolyte type (LiCl, NaCl, KCl, CsCl, CaCl2, SrCl2, and BaCl2). Interpretation in terms of the surface conductance revealed that a substantial part of the surface conductivity originates in the inner part of the double layer. Electrokinetic potentials and related diffuse double layer properties are highly dependent on the nature of monovalent counterions, whereas divalent counterions do not show such clear dependencies. Further presented was a simple way to estimate the order of magnitude of counterion mobilities in the inner part of the electrical double layer. All counterions were shown to have a substantial mobility in the inner part of the double layer. Finally, we suggest that the apparent ion-specific effects observed in the diffuse part of the double layer are at least in part related to the finite size of the counterions. Our findings are relevant to scenarios where fluid flow in porous media is accompanied by charged species transport, e.g., in electro-osmotic remediation, spectral-induced polarization, or permeability measurements.