While there are quite a few fictional languages in the books and in the TV series, George R.R. Martin said that he never intended to create full languages like J.R.R. Tolkien did for Middle Earth. So that is why, for the TV show, a linguist was hired in order to create these artificial languages.
So it isn’t quite a surprise that some of the words and expressions have a catch to them and they sound so natural, or that they have such resonance. And the best example to be given in here is Valar Morghulis – and everybody knows that this means that all men must die.
But as everybody knows, everything that we imagine, no matter how eccentric it might seem, is based on reality, on what we have experienced until now. Such is the case of this expression, whose meaning can be traced back to another celebrated saying: Memento Mori.
In both cases, the reader is made aware that the same fate awaits all men: Death hovers above us all and we don’t know when it will strike us. Furthermore, we see that this particular expression (Valar Morghulis) is uttered by Arya when giving a coin to the captain of the boat.
As far as I am concerned, this seems to be not just a coin with which a Faceless Men can be recognized, but also a symbol coming from the Greek mythology. In this case, the captain is a symbol himself. He is Charon, the one that carries the souls of the dead across Styx and Acheron, the rivers which separate the land of the living from the land of the dead.
These are just speculations for now, but these speculations follow closely what we have seen in the Game of Thrones series. And, in my opinion (without any reference to the books), it is pretty clear that Valar Morghulis is in fact an incantation with which the coin is recognized.
Arya is a tomboy in the beginning of the series and she is quite lovable the way she is. However, we see her changing as the plot develops, until a moment when we see her actually killing without any remorse. Even more, her changing becomes obvious when she refuses to help the Hound (although he has helped her in numerous occasions).
In other words, in my opinion, Valar Morghulis signifies the fact that Arya is prepared to go on the other side completely. If she has changed until now (until the death of the Hound), the fact that she pays the captain his coin means that her change is irreversible.
Of course, we will still have to wait and see if indeed all men must die, if indeed Arya Stark has passed well beyond the point of no return.