1. She is beautiful, and not only that, she is also the most beautiful being in Middle-Earth.
While some might say this is a stupid reason to love a character, I don't agree with them. I'm just too stunned by her beauty and I also think she's beautiful on the outside because she's beautiful on the inside.
2. She is wise and smart
When you've lived over 2,700 years in Middle Earth, you've seen and lived a lot of things. You can think better and make better decisions based on what you know. I always admire people's knowledge and Tolkien tells us she has all the wisdom and knowledge that those years can bring. She knows what she has to do and when, she knows that if her daddy tells her to come back to Rivendel during the war of the Ring, she'd better do what he says ^_^ because ke knows best. Also, as Elrond's daughter, I'm sure she's had a great education: Elvish sorcery, warfare and scholarship. She must know a thing or two about horses and she's very good at handicrafts (the banner she made for Aragorn, ge), which can be very useful if, for example, you have to sew an Elven cloak like the ones Galadriel made for the Fellowship. She also knows how to cut gems, and she must have practised some healing arts. Well, she must know some more things, I suppose :D
3. She is no warrior
Strange reason to love a character, isn't it? but yea, I love her because of this, too. I reckon she is a capable woman, that she can defend herself. I like to think that, even if Tolkien never says it literally. Thing is, I'm really tired of the warrior female sterotype, it's becoming boring and non-warrior characters give us a differente glimpse of the situations, especially when it's a war. I liked the fact that she can do other things rather than go into battle to kill orcs, like her brothers do. And maybe because she isn't a warrior, she deals with her mother's capture differently than her brothers. She doesn't panic and go outside to kill as many orcs as she can. She's more philosophical, so to speak.
4. She is compassionate
According to Tolkien, Arwen was the first one to notice that Frodo would never recover from his wounds, so she thought of a way of healing him. Then, she gave him a jewel to give him comfort and gave up her place on the ship to Valinor for him. I don't think she's ever spoken to him, she probably knew him thanks to Bilbo, whom I'm sure she talked to, but anyway, she knew his fate and immediately thought of a way to help him.
Sometimes I've wondered: what would have happened if Arwen had met Eowyn during the war of the Ring? and the obvious answers was: she would try to help her somehow. I know that 100 % sure and it's great. I love her for being so nice to other people.
5. She IS STRONG
Nowadays, it's like there can't be a strong female character if she isn't a warrior, a physical warrior, I mean. But Arwen is strong. First, she had to face her mother's capture and torture and then she had to say goodbye to her, with the possibility that maybe she wouldn't see her again. Then she met Aragorn, and they fell in love and she had to make a very difficult decision: to go to the Undying Lands with her father and the rest of her kin or to stay in Middle-Earth with Aragorn, sharing a mortal life with him, which would mean she would never see her parents again, not even in the after life. People tend to underestimate this, since death is the gift of Ilúvatar to Men, but it's still a very difficult thing to do. First, because it separates her from her family forever. Secondly, nobody knows what's after death, not even the wisest of the Elves know that for sure, though most of them seem to belive that there is nothing, that Men cease to exist. So imagine Arwen, when Aragorn dies and she has to face the fate she chose herself. Also, sometimes I think about the moment when Elrond and Arwen are in Rohan and they say farewell to each other, with nobody watching them, it must be so hard. I don't think I could do such thing. And she was also strong enough to remain in Rivendell while Aragorn went to face his destiny (though I know she had foresight and she knew he would get his goal)
6. Yes, she does something during the War of the Ring.
To begin with, just because Tolkien doesn't tell you something, doesn't mean it didn't happened. So though many people believe she only sit and waited for everything to finish, I'm sure she had to do something else. I don't think she is so passive. And then there's the banner she made for Aragorn. It's large, it's dificult and she made herself, nobody else. Also? Read this article: http://alexbeecroft.com/2011/08/in-defence-of-arwen/ After I've read it, I think Arwen has a lot to do in Aragorn's victory.
7. She isn't perfect, she has flaws.
I read somewhere that Tolkien wrote in a letter (non published) that Arwen could have chosen to die right after Elessar did, but she wasn't ready yet. She was scared by her own death because, as I've said before, nobody knows what's after death. See? she wasn't perfect. I like the fact that, even at that point of her life, she learns something. She had scorned the Men of Numenor because they feared death and at that moment, she understood them. She, of all Elves in Middle-Earth, understood it and empathized with them. How can you not love her?
If you have anything else to add, you're most welcome :)