You may (or may not) be able to get the name you want - it depends on what process you start.
Process p = new Process();
p.StartInfo.FileName = @"D:\Temp\MyText.txt";
p.Start();
Console.WriteLine(p.ProcessName);
Will almost certainly give you the process name "Notepad" (or in my case "PsPad") -
but it may throw an "invalid operation" exception if you open (for example) an HTML file. On my system, that will open with Chrome, but since Chrome throws the request to an already running instance and terminates the process immediately so as not to generate a new window it doesn't work (because the Process ID is no longer running).
Still, it might be what you want, and it's pretty much the only way to get it...