For this, we need to go back a lot - ages back - and examine Arwen's parentage, and the Lay of Leithian, the story of Beren and Lúthien. This is a rather long story (the longest in the Silmarillion, I believe, or perhaps the second longest after the Narn i Chîn Húrin, the story of the Children of Hurin), and while I strongly urge you to go read the Silmarillion, I'll summarize it briefly:
Way back, before the time of Lord of the Rings and The Hobbit, in the First Age of Middle Earth, there was an Elven woman, Lúthien, the daughter of Elu Thingol, King of Doriath, and one of the three great lords of the Elves in the very beginning. She fell in love with the human Beren, and despite much tragedy that would put Romeo and Juliet to shame, Beren dies (since he is human), and Lúthien petitions Mandos, the Valar in charge of the fates of all mortals, and she gives away her immortality so that they could live together, for as long as they have. This is the first turning point.
This is a very tragic tale, of course. The Elves were devastated over it, since Lúthien was the fairest and most beautiful maiden to ever live, etc, and once she became human, she was destined to die, and pass away, and her beauty seen no more, and so forth. It's sung as the greatest sacrifice made for love, etc.
Now, from this tragic romance we have a son of Beren and Lúthien, one Dior Eluchil. He married Nimloth, a Sindarin elf, and had a daughter, Elwing. She married Eärendil, who was also half elven, with his own tragic backstory linked to the Children of Hurin. They had two sons, Elrond and Elros, who were just children when the First Age ended with a great and terrible war. Then, we come to the second turning point. The two children, for the bravery of their parents and the suffering they underwent, were given the choice whether to be counted among the Elves or among Men. Elros chose Mankind, became the King of Numenor, and from him were descended all the Kings of Men, down to Gondor. Elrond, however, chose to be an immortal Elf, founded the haven at Rivendell, and had his daughter, Arwen. (Yes, that does mean Arwen and Aragorn are related, but very distantly).
So how does that relate to the question? Well, Arwen is the descendant of many Elves and Humans (and a bit of Maiar, but that is irrelevant) who have fought bravely against Morgoth and Sauron, and were given the choice, due to their mixed parentage and Luthien's sacrifice, of where to be counted. Arwen simply inherited that with the rest of her respectable ancestry.