I see a lot of books being listed here that, while they might be "good" books, I'm not sure how they're actually "influential" books. Since "influential" is what you asked for, I will attempt to pare my list down to only those books which had a true life and/or perspective altering effect for me.
Most of the classics have been mentioned here already, so I'll leave those off the list, and just mention this one:
The Mists of Avalon by Marion Zimmer Bradley.
This is an amazing book, and one that I regularly revisit. Basically, it takes the premise that history is written by the victors and applies it to Arthurian legend, which was originally penned by Christian monks. It examines what the other side of the story might be, by telling it from the point of view of Arthur's sister, the "sorceress" Morgaine. What if she hadn't been a sorceress--what if she had merely been a priestess of the old pagan worship that was being condemned as the work of the devil and swept away as Christianity became the dominant religion in the British isles. What if her struggle with Arthur was not to undermine his throne and claim power for herself, but to preserve her way of life and worship?
The book addresses how some of the more commonly known folk-tales surrounding Arthur's reign were born--such as the story of the sword in the stone and the gifting of Excalibur to Arthur by the Lady of the Lake, and how he came to sleep with his own sister and father a son by her. It also takes the occasional moment to poke fun at the way common tales get inflated in folklore, such as when the Bishop Patricius brags about having stomped out the druidic worship in Ireland (the druids being known as the Serpents of Wisdom) and thus gives rise to the tale that he drove the snakes out of Ireland.
It's a wonderful novel that really turns on its head the "good vs. evil" paradigm that traditionally infuses Arthurian legend and instead makes it about the struggle of an old way of life against the rising tide of a new way.
Along those same lines, though with much less impact, I would also recommend Phantom by Susan Kay. This is the story of the Phantom of the Opera, as told in first-person perspective from people who had contact with Erik, the disfigured man who would become the Phantom. It begins with his mother from his birth and through his early childhood, then is told for a while by Erik himself, during his adolescence as a side-show freak displayed by the Gypsies. It moves on to his time in Persia from the point of view of the man referred to in the original Gaston Lereaux version as The Persian, and then again from Erik's point of view during the building of the Paris Opera House. The saga we all know involving Christine Daae is told counterpoint between Erik and Christine, and the conclusion is told by Raoul, the Viscount de Chagny.
Like The Mists of Avalon this book takes an old story with which we are all familiar and turns it on its ear, approaching it from a new perspective.
I think I'm detecting a trend in my preferred reading material...
Edited, Jan 2nd 2007 9:59am by Ambrya