The determination of the solvation free energy of ions and molecules holds profound importance across a spectrum of applications spanning chemistry, biology, energy storage, and the environment. Molecular dynamics simulations are powerful tools for computing this critical parameter. Nevertheless, the accurate and efficient calculation of the solvation free energy becomes a formidable endeavor when dealing with complex systems characterized by potent Coulombic interactions and sluggish ion dynamics and, consequently, slow transition across various metastable states. In the present study, we expose limitations stemming from the conventional calculation of the statistical inefficiency g in the thermodynamic integration method, a factor that can hinder the determination of convergence of the solvation free energy and its associated uncertainty. Instead, we propose a robust scheme based on Gelman-Rubin convergence diagnostics. We leverage this improved estimation of uncertainties to introduce an innovative accelerated thermodynamic integration method based on the Gaussian Process regression. This methodology is applied to the calculation of the solvation free energy of trivalent rare-earth elements immersed in ionic liquids, a scenario in which the aforementioned challenges render standard approaches ineffective. The proposed method proves to be effective in computing solvation free energy in situations where traditional thermodynamic integration methods fall short.