The GENERIC structure allows for a unified treatment of different discrete models of hydrodynamics. We first propose a finite volume Lagrangian discretization of the continuum equations of hydrodynamics through the Voronoi tessellation. We then show that a slight modification of these discrete equations has the GENERIC structure. The GENERIC structure ensures thermodynamic consistency and allows for the introduction of correct thermal noise. In this way, we obtain a consistent discrete model for Lagrangian fluctuating hydrodynamics. For completeness, we also present the GENERIC versions of the Smoothed Particle Dynamics model and of the Dissipative Particle Dynamics model. The thermodynamic consistency endorsed by the GENERIC framework allows for a coherent discussion of the gas-liquid phase coexistence of a van der Waals fluid.