actually the script works for me in the following context.  firstly you are missing a semicolon at the end of one of your first few lines.  secondly i am using the name symbolItemName as a variable instead of symbolItem.  i am also using aics6

#target illustrator
 
symbolSwapper()
 
function symbolSwapper() {
  
          var doc, count, red, blue, i;
  
          doc = app.activeDocument;
  
          count = doc.symbolItems.length;
  
          red = doc.symbols.getByName( 'Red Circle' );
  
          for ( i = 0; i < count; i++ ) {
  
                    doc.symbolItems[i].symbol = red;
  
          }
  
          app.redraw();
  
};

var docRef = activeDocument;

// your previous variable name may have been a keyword.  also, did you check in your symbol gallery whether the name was // changed to "red_circle"?

var symbolitemName = docRef.symbolItems[0];

var symbolname = symbolitemName.symbol.name;

alert ("the symbol's name is " + symbolname)

// this can be anything that is not being used as a keyword

var symbolNewName = "red_circle";

symbolitemName.selected = true;

// this next line causes a problem:

symbolitemName.symbol.name = symbolNewName;

I have verified the following:

1) Muppet Mark's script actually changes the symbol due to his use of the app.redraw() function in addition to the other bits of code he used leading up to that.

2) Your script when properly syntaxed (with variable names not conflicting with keywords) changes the name of the symbol but does not change the symbol that is on the stage.

In other words in the first case you have drawn a blue square and a red circle onto the stage and converted each of them to a symbol (Movie Clip is fine) with the name "blue_square" and "Red_Circle".  Note the underline in Red_Circle which you will have to amend in Muppet Mark's script in order not to get any errors.  Delete the original symbols from the stage as they are in your library and drag new ones to the stage from your symbol library, one of each.  give them instance names (I don't know if this step is optional or not).  You will see that you now have two red circles on the stage after running Muppet Mark's script.

In the second case you do exactly as the first case but you will only notice that the symbol name "Red_Circle" changes to "red_circle" when running your script but no symbol instances are changing on the stage because no app.redraw() is being called.  Perhaps the nature of the script might not allow for any redraw anyway but I can verify that both scripts work and do what I have described in the previous two paragraphs.

You will see that I have wrapped all my syntax in a function… This is good practice with scripting AI as the engine is persistent throughout the session… Declaired all my variables at the off… inside the function wrapper. I didn't bother to check that a document was open nor try/catch the getByName() which you should… I knew my doc was open and contained what I wanted…