The best materials for a running hat depend on the weather conditions that you’ll typically run in. If you expect to run in winter, or in especially cool or wet weather, you’ll want to look for a hat that has solid material instead of mesh, a lining, and possible reflectivity for the darkness.
If you typically run in the summer, or it hot and dry weather, you’ll want a hat that is lighter weight, has moisture-wicking materials, mesh construction or other ventilation features, and UPF protection.