In this paper we develop a methodology for the mesoscale simulation of strong electrolytes. The methodology is an extension of the fluctuating immersed-boundary approach that treats a solute as discrete Lagrangian particles that interact with Eulerian hydrodynamic and electrostatic fields. In both algorithms the immersed-boundary method of Peskin is used for particle-field coupling. Hydrodynamic interactions are taken to be overdamped, with thermal noise incorporated using the fluctuating Stokes equation, including a "dry diffusion"Brownian motion to account for scales not resolved by the coarse-grained model of the solvent. Long-range electrostatic interactions are computed by solving the Poisson equation, with short-range corrections included using an immersed-boundary variant of the classical particle-particle particle-mesh technique. Also included is a short-range repulsive force based on the Weeks-Chandler-Andersen potential. This methodology is validated by comparison to Debye-Hückel theory for ion-ion pair correlation functions, and Debye-Hückel-Onsager theory for conductivity, including the Wien effect for strong electric fields. In each case, good agreement is observed, provided that hydrodynamic interactions at the typical ion-ion separation are resolved by the fluid grid.