Accurate high-resolution wave forecasts are essential for coastal communities, but local and even coastal coverage is often still missing due to the heavy computational load of modern state-of-the-art wave models. This study presents a machine learning super-resolution approach that drastically reduces the computational effort, while keeping errors negligible for the majority of forecasting applications. The method consists of first computing a wave forecast on a coarse mesh which is then converted to a forecast of finer resolution with the help of machine learning. To demonstrate the feasibility and the potential for practical applications of this approach, we present a case study of a 44-year hindcast along the French Basque coast over an unstructured mesh. We introduce two machine learning approaches, a graph neural network and a polynomial ridge regression and compare their performances in different sea states and spatial environments. Both models exhibit very small prediction errors for the significant wave heights, with Root Mean Square Errors (RMSEs) ranging from 0.3 cm to 2 cm, depending on the study region, while being up to 80 times faster than a direct computation of a numerical wave model at the corresponding spatial resolution. To the best of our knowledge, this is the first time that a super-resolution approach is extended to unstructured meshes in the field of coastal sciences.