Mathematical models of biochemical reaction networks are an important and emerging tool for the study of cell signaling networks involved in disease processes. One promising potential application of such mathematical models is the study of how disease-causing mutations promote the signaling phenotype that contributes to the disease. It is commonly assumed that one must have a thorough characterization of the network readily available for mathematical modeling to be useful, but we hypothesized that mathematical modeling could be useful when there is incomplete knowledge and that it could be a tool for discovery that opens new areas for further exploration. In the present study, we first develop a mechanistic mathematical model of a G-protein coupled receptor signaling network that is mutated in almost all cases of uveal melanoma and use model-driven explorations to uncover and explore multiple new areas for investigating this disease. Modeling the two major, mutually-exclusive, oncogenic mutations (Gαq/11 and CysLT2R) revealed the potential for previously unknown qualitative differences between seemingly interchangeable disease-promoting mutations, and our experiments confirmed oncogenic CysLT2R was impaired at activating the FAK/YAP/TAZ pathway relative to Gαq/11. This led us to hypothesize that CYSLTR2 mutations in UM must co-occur with other mutations to activate FAK/YAP/TAZ signaling, and our bioinformatic analysis uncovers a role for co-occurring mutations involving the plexin/semaphorin pathway, which has been shown capable of activating this pathway. Overall, this work highlights the power of mechanism-based computational systems biology as a discovery tool that can leverage available information to open new research areas.