Please can anyone explain what is the "fight club theory" in Game of Thrones? And does this theory relate to the books in any way?
I hadn't heard of this theory, but found it by Googling "fight club theory" "game of thrones".
From TV Guide's Game of Thrones: All the Crazy Arya Theories, Explained:
The Fight Club Theory: Some fans suggest that Arya and the Waif are not actually nemeses, but the same person, with the Waif being the Tyler Durden to Arya's Edward Norton. The meat of this theory is that the Waif and Arya represent her two possible paths - becoming No One and embracing her born identity - and that all the fights between Arya and the Waif are her inner struggles literalized. That would make the Waif's attack on Arya a (failed) last-ditch effort to kill her emotions and officially become No One.
However, given the absurdity of this theory and the fact that the Waif is equally emotional as Arya, it doesn't quite line up.
This is a reference to the 1999 film Fight Club, and if you read Wikipedia's plot description you'll see why.
The theory may have originated on Reddit last Tuesday (31 May 2016).
As for whether it relates to the books, you have to remember that A Song of Ice and Fire and Game of Thrones have considerably diverged at this point, and the show has moved beyond all the currently existing books. It may be that the theory will also be valid in the books; it may also be that it will be invalidated even in the show when the next episode airs.