"No fear upon that way was so present as the fear for what might befall her." -Aragorn in the Houses of Healing
Why Aragorn and Éowyn? It's difficult to explain point by point, besides simply saying, "There's something so right about it." They have so much chemistry in the book, especially in passages like the one where Aragorn sees her as a glimmer of white, with her eyes on fire. It's almost like Tolkien wanted us to feel this energy between them, and then pulled the rug from under our feet. Or perhaps he couldn't get his original intent fully out of his mind, and Aragorn and Éowyn connected in spite of the redirected plot. And the fact is, they do connect.
I feel very strongly that Aragorn needs Éowyn. He's spent decades as a wanderer and Ranger, and his heart clearly lies in the freedom and adventure of the wild. When he accepts his duty to become king of Gondor, he sacrifices a great deal - including the kind of life that makes him happy. With Éowyn, he would have someone who understands his heart and loves it for what it is. He would be removed from the wild, but he would have someone who makes life just as exciting and adventurous. And if he ever goes wandering in the wild sometimes, which I'm sure King Elessar did, he would have a wife who goes gladly with him and enjoys herself.
I know it's highly debatable that Éowyn loved Aragorn, but I think she did. Many argue that she just had a crush, but I think this was a woman who was not capable of a crush, emotionally. Her heart had turned cold, and all around her she saw despair. For Aragorn to come into her life and get through to her, I think she loved him. Of course, it was only the first stage of love - or she couldn't have turned so quickly to loving Faramir. But this was no crush. People do not beg to ride to death with their crushes.
Others say that she loved only the idea of Aragorn. This argument does make good sense to me, even though I disagree with it. I think that she initially fell in love with what he stood for, rather than with Aragorn himself. She saw him as a kind of savior, but soon fell in love with Aragorn himself. After all, she must have seen every good quality in Éomer - so Aragorn wasn't a kind of man she had never met. It is argued that Éowyn only saw Aragorn a couple of times, and they only had one conversation, but that is incorrect. When Éowyn gives Merry his armor, she mentions part of a conversation with Aragorn that doesn't appear in the book. Evidently they did interact in scenes that Tolkien didn't give us.
As for the argument that she just wanted the power, that's ridiculous. She was practically a queen already - she was even given command of Rohan, and she chose to ride with the army instead. Tolkien himself, in a letter to a reader, said that Éowyn did not love Aragorn because of political ambition. Finally, Éowyn didn't even find out until later that Aragorn was a king.
Éowyn loved Aragorn for who he was. Arwen saw only a future king, and refused to marry Aragorn if he wasn't a king. Don't give me the "obeying her father" argument... we're talking about waiting over forty years. Lúthien defied her father for Beren, and nothing held Arwen back but her own refusal to be with a Ranger. In fact, if anyone loved only "a shadow and a thought," it was Aragorn's instant love for Arwen.
I do think that Aragorn loved Arwen. BUT... I also firmly believe that he was attracted to Éowyn, even though he knew he couldn't act on it. There are textual hints of it. In a few places when Tolkien describes Éowyn, he specifically mentions that Aragorn is the one watching her, even from a distance. In the Houses of Healing, referring back to the Paths of the Dead, Aragorn even goes so far as to say: "No fear upon that way was so present as the fear for what might befall her." With all he had to fear - all the concerns pressing his mind - his first thought was for Éowyn. He cared about her deeply. Loved her. Had circumstances been different - had there been no Arwen - Aragorn would have married Éowyn. I'm sure of it.
I've heard the argument that Aragorn is too old for Éowyn, and that one makes me laugh, considering that Arwen is thousands of years older than he is. But I'll be serious, and address this more from an angle of maturity. Yes, Aragorn was old and had seen a great deal of the world. But Éowyn was no dumb blond. The men and king of Rohan obviously respected her so much that they were willing to let her rule - and let's not forget that this is a primitive society. For them to grant such a high position, such respect, to a woman, is saying a great deal about Éowyn. Éowyn was perfectly capable of being an intellectual and emotional match for Aragorn.
And she was his match, as he was hers.
In the end, Tolkien rejected Aragorn/Éowyn. Initially, he planned for Éowyn to die, and for Aragorn never to marry after her death. But then he wrote Faramir and Arwen, and things ended - at least for one pairing - happily ever after.