I never liked that phrase either. So what exactly lives in your head and pays rent?
living rent free generally means that you are able to easily recall a piece of knowledge (and often do, even if you don't want to).
something that pays rent might be like an important concept for an upcoming school test, for a class you're not interested in.
The term in its original use described an obsession with a particular group or thing and called it out as pointless. I don't think it would make sense for rent to mean effort in this original sense, nor in the broader way it's used here.
In addition, it literally says "rent free". There's nothing intellectual or pseudo-intellectual about reading the phrase "rent free" and concluding that the information does not pay anything to live in your head. This is basic reading comprehension.
Get a hobby.