A project's name is its web address, for example www.example.com: SyncNow, the agents and the managers know the project by it, so it does not change after the project is created.
Project settings shows the name as the Project address but does not let you edit it, and the Connections page changes only how SyncNow reaches the site (FTP server, folder, password and databases), not the project's name. If the hosting moves to another server, change the hosting connection: the project keeps its Planning, Rules and Activity.
If the site itself moves to a different web address, add it with New project as a new project; the old one stays in your list until you remove it. See Where are a project's settings, and what can I change there?.