A script to add the illustrator file name, minus extension, to a text field within the file.

So you close the file and rename it. You open the file again, and you want the text frame to update automatically? Scripts only change the document when you run the script.

If you have a text object with a key name in the file, you will need to run the script each time to insert the filename into it. Example:

(function (){
  var doc = app.activeDocument;
  var docName = doc.name.replace(/\.[^\.]+$/, '')
  var targetName = "filename";
  try {
    var target = doc.textFrames.getByName(targetName);
    target.contents = docName;
  } catch (error) {
    alert("Text frame '" + targetName + "' not found");
  }
})();