Many of the story elements Game of Thrones changed from A Song of Ice and Fire were done for the sake of simplicity, time or to take the story in a new direction. Multiple characters were composited into single individuals, or subplots were excised altogether to make room for other content in the story. However, when making its changes, Game of Thrones rarely opted to leave out its coolest portions. Dragons and fight scenes stayed or were given extended roles, except where the Faceless Men were concerned. Despite being one of the coolest elements of its world, Game of Thrones introduced the shapeshifting assassins without ever really knowing what to do with them. In fact, some of their best subplots from the books were left on the cutting room floor.
At first only hinted at through whispers and rumors, the Faceless Men of Braavos are built up in Game of Thrones as a mysterious but titillating force. The small council considers hiring them early on to execute Daenerys, but balk at the expense. Such high-priced assassins were sure to catch viewers' interest, but they became only more engaging when Arya ran into one.
After freeing Jaqen H'ghar from a prison where he nearly burnt to death, Arya earned herself the favor of the assassin, who swiftly proved capable not only of killing anyone she named, but doing so in a way which appeared accidental. When Jaqen left Arya, he did so with an iron coin and instructions on how to find the Faceless Men if she ever needed them before changing his face and slipping away.
It was a fantastic introduction, and it quickly tied the Faceless Men to Arya's story so centrally, but the organization never served any purpose in the story beyond providing Arya with the skills she needed to enact her revenge. When she went to Braavos, she trained with the Faceless Men, but she secretly held on to her personal identity despite their repeated instruction she abdicate herself to the Many Faced God they served. In the end, she succeeded in fleeing Braavos with her newly won skills intact, leaving the Faceless Men behind to never have further consequence in the story. Looking to the books where their subplots hint at something far greater, this may have been a tremendous missed opportunity.
While retaining their mystery, further insights into the Faceless Men of A Song of Ice and Fire manage to only ever make them more interesting. One key plot point that has yet to come to fruition in the series involves one Faceless Man, implied to be J'aqen following his encounter with Arya, arriving at the Citadel for an unknown purpose. The Citadel is the central point where maesters are trained and is one of the most vast collections of knowledge in the world, and the Faceless Man disguises himself as a young novice who gains access to a master key that unlocks every door in the Citadel. Such designs hint at a grander scheme on the part of the Faceless Men.
Even Arya's training in Braavos proves notably different in the books. In the show, Arya is stricken with blindness as a punishment, but the books portray the blindness as a natural progression of her training. One of the larger story elements for Arya that Game of Thrones downplayed was her status as a warg, capable of inhabiting the minds of animals, and in the books, she learns to warg into a nearby cat in order to regain her sight. The Faceless Men also utilize Arya for gathering information around Braavos, and she undergoes missions like the assassination of a Brother of the Night's Watch who abandoned his post.
If anything, the Faceless Men of the books do not seem to merely be training Arya for the simple satisfaction of gaining a new member, but rather as an extension of further, unseen goals with greater ambitions behind them. Arya is after all a key figure in the world, as not only one of the few remaining Stark children but one of the few wargs learning to hone their abilities. It only makes sense that an elite cadre of assassins would want to use her as much as she would want to use them.
Mysteries are only ever really satisfying when they have substance to their ending. Anyone can raise a puzzling question, but finding a clever answer is far harder. Game of Thrones made the Faceless Men a fascinating mystery that never really resolved into anything substantial, and it may just be up to the books to give fans the clever answer they crave.