I would like the MP!L popup to display the contact's nickname rather than their login name.

Is there a variable or code I can put in the custom message section to show the nickname?

I'm sorry but this is not possible.  Currently, due to MP!L being recoded, there is no custom names feature (this may not even return at all, it is Patchou's decision) and there has been no way so far for anyone to find a method to get the custom nickname you can give you contacts through WLM's options.  Because of both these facts there is no way of doing such a thing I am sorry to say.  The only way would be if someone were to make a script where you can assign a nickname to your contact and run all your toasts through that script, but to do something like that is a massive effort and I do not think any scripter will offer to do such a thing.

In short, no there is no variable or anything, this cannot be done at the present time and may never be done in the future either (lets hope otherwise however).