Solid-liquid phase diagrams are calculated for binary mixtures of Lennard-Jones spheres using Monte Carlo simulation and the Gibbs-Duhem integration technique of Kofke. We calculate solid-liquid phase diagrams for the model Lennard-Jones mixtures: argon-methane, krypton-methane, and argon-krypton, and compare our simulation results with experimental data and with Cottin and Monson's recent cell theory predictions. The Lennard-Jones model simulation results and the cell theory predictions show qualitative agreement with the experimental phase diagrams. One of the mixtures, argon-krypton, has a different phase diagram than its hard-sphere counterpart, suggesting that attractive interactions are an important consideration in determining solid-liquid phase behavior. We then systematically explore Lennard-Jones parameter space to investigate how solid-liquid phase diagrams change as a function of the Lennard-Jones diameter ratio, sigma(11)/sigma(22), and well-depth ratio, epsilon(11)/epsilon(22). This culminates in an estimate of the boundaries separating the regions of solid solution, azeotrope, and eutectic solid-liquid phase behavior in the space spanned by sigma(11)/sigma(22) and epsilon(11)/epsilon(22) for the case sigma(11)/sigma(22)<0.85.