Lattice models are crucial for studying thermodynamic properties in many physical, biological and chemical systems. We investigate the lattice restricted primitive model (LRPM) of electrolytes with different discretization parameters in order to understand thermodynamics and the nature of phase transitions in systems with charged particles. A discretization parameter is defined as a number of lattice sites that can be occupied by each particle, and it allows one to study the transition from the discrete picture to the continuum-space description. Explicit analytic and numerical calculations are performed using the lattice Debye-Hückel approach, which takes into account the formation of dipoles, the dipole-ion interactions and correct lattice Coulomb potentials. The gas-liquid phase separation is found at low densities of charged particles for different types of lattices. The increase in the discretization parameter lowers the critical temperature and the critical density, in agreement with Monte Carlo computer simulation results. In the limit of infinitely large discretization our results approach the predictions from the continuum model of electrolytes. However, for the very fine discretization, where each particle can only occupy one lattice site, the gas-liquid phase transitions are suppressed by order-disorder phase transformations.