In most cases, the name of a PhpStorm project is the same as the name of its root folder, so the easiest way to change the name of a project is just rename its root folder.
In this case however you have to re-configure access to the server so that the server root is mapped to the new project root name. To avoid these additional steps, you can change the name of the project but have its root folder name preserved. PhpStorm in this case will only modify the project metadata.
Right-click the root folder of your project and selectfrom the context menu or press Shift+F6.
In the dialog that opens, choose the rename strategy.
If the project name is the same as the name of its root folder, select Rename directory. PhpStorm will perform the Rename refactoring so that all the references to the directory throughout the code remain valid.
If the project name is different from the name of its root folder, select Rename project. Alternatively, you can select from the main menu and in the dialog that opens, type the new name of the project. Also select this option if your application is deployed to a remote server and the project root folder is mapped to the server root.
Of course, you can rename the root folder and update the mapping in the server configuration accordingly, see Configuring synchronization with a web server for details.