Expressions that have no meaning (什么/来着） I know that these two expression, in certain circumstances, have no meaning and are optional. I'm trying to figure out how to use them, even though they have no meaning.

什么

This has a meaning, (negation):


  大家都是朋友，你客气什么呢。


On the other hand, this use of 什么 means nothing:


  我没什么说的了。


来着

This has a meaning (forgetting something you should remember):


  怎么写这个字来着？


This use of 来着 has no meaning:


  你的女朋友到底是谁来着呢？


Just to clarify, I understand the situations where these expressions have a defined meaning. I just can't get my head around the usage when the expression is optional. I can leave it out, and I'm content with that. But for some reason my teacher in Beijing felt the need to show this to me, maybe to make me sound less 'academic' when I talk?
I am a native speaker, and I'm no expert. But I do think you have some misunderstanding towards "什么" and "来着" in these examples.


  大家都是朋友，你客气什么呢。


In this sentence, "什么" means "what ... for". The whole sentence means something like: "(Considering) we are friends, (I have to ask) what are you being 客气 for?" You can say it helped in expressing negation. (sorry I can't translate 客气 accurate enough)

But in the second example


  我没什么说的了。


"什么" doesn't mean "nothing". On the contrary, it means "anything". And "没什么" as a whole, means "nothing". The whole sentence means "I don't have anything to say."

As for "来着", I think it means exactly the same (forgetting something you should remember) in both of your examples. And I can't think of any cases in which this word means anything else or has no meaning at all.

So to sum up, I don't think there is any cases in which these words has no meaning at all, at least not in your cases.

btw. "你的女朋友到底是谁来着呢？" should really be "你的女朋友到底是谁来着？" (No "呢" at the end)