None of the answers mention writing a unit testable solution. You could use System.IO.Abstractions
as it provides a testable wrapper around FileSystem operations, using which you can create a mocked file system objects and write unit tests.
using System.IO.Abstractions;
IFileInfo fileInfo = _fileSystem.FileInfo.FromFileName("filePathAndName");
fileInfo.MoveTo(Path.Combine(fileInfo.DirectoryName, newName));
It was tested, and it is working code to rename a file.