Eowyn was a contrast to Arwen. Eowyn was a young woman, Arwen was a 500 year old elf queen. Arwen was described as Tinuviel coming back to life. That's like saying, "She looks like Eve!" Eowyn was described as a young maiden, hair like a river of gold, and not yet in full bloom (small breasted). Arwen loved and felt compassion for Aragon, especially concerning human mortality. Eowyn's love seemed colder and more sympathetic for Faramir, especially concerning his wounds and depression. Arwen married downward (according to Elf standards) and Eowyn married someone of equal status in a neighboring country.
Arwen's involvement with Mordor was political; she didn't get personally involved. When Aragorn was in trouble, she sends a sword for glory, a gem for recognition, and some people to fight with him. Eowyn got personally involved and slew Sauron's right hand man, with Merry's help. I especially liked what each woman did with their swords; Arwen sent Narsil (the sword) to Aragon through her father while Eowyn hid hers under her saddle blanket. It indicated their diverse personalities. If you upset Arwen, she'd send someone to kill you. If you upset Eowyn, she'd kill you herself!
Eowyn's character carries her through the story. Eowyn was a typical tomboy and trained swordswoman. Her uncle and only remaining guardian seemed to fall ill (due to Wormtongue's influence). The country wasn't as well-run, her uncle wasn't recovering, and her cousin and her brother ran off to battle. The boys coped by running away to fight orcs. Eowyn, being a woman, couldn't run away to do battle. Eowyn suffered in silence, watching her once proud, tall, and strong uncle dwindle into a withering, stunted tree.
She lost her confidence. Wormtongue, her supposed counselor, made her feel worse. Her cousin's death was devastating: she might lose the kingdom and her uncle. Eowyn, however, had nerves of steel beneath her. She hid a sword beneath her saddle blanket to go to war to steel her nerves, but she still didn't have enough confidence. Eowyn hung on to Aragorn, who had the confidence, strength, and honor as king that she had lost as a princess. He didn't like her fiesty response in an argument or hiding her sword under a saddle blanket (symbol of hiding her strength under a blanket of deception), so he stuck with Arwen.
Eowyn went into battle to confront her fear: that the enemy was unstoppable and her uncle would die. Her fear came true; her uncle died. The enemy, Sauron's right hand man, told her "No man can kill me," then Merry stabbed his sore spot behind the knee, and the enemy screamed his head off. This told her, "If he can feel pain, if he's made of flesh like you, he's stoppable." With her confidence back, she was proud to be a swordswoman and sticks her hidden sword right through Sauron's servant's head!