Roleplaying

Eowyn's role in The Two Towers is tiny, but she's there; during the first half of the game, all the Fellowship of the Ring-timed quests are done in flashback mode, with Aragorn telling Eowyn of his deeds before he ended up in Rohan. She's not seen until the second-to-last mission in the game, where she tells Aragorn that "The women and children are being trapped!" And...yeah. That's about it.

She has a bigger role in Return of the King, but sadly enough, she's still not playable. The one time you see her is during The Battle of Pelennor Fields. After you kill 60 Easterlings or Orcs, she and Merry show up on the battlefield. Your job is to kill the mumakil that are threatening them, and keeping their health bar up. You can do this quest as either Aragorn, Legolas, and Gimli; since Legolas has the strongest ranged attack of the three, you might want to fight as him for this one. Shoot the Mumak to bring down their armour, then target the explosives its carrying. After you kill two, the witch king swoops over Eowyn and Merry. Now your job is to target the Fell Beast the Witch King is riding. When you do enough damange and the witch king swoops out of range, another Mumak shows up. Once you take care of that one, the witch king comes back to harrass Eowyn and Merry. Your task here is to keep Eowyn and Merry safe, and they can only be harmed by the Witch King and by Mumakil; therefore, they should be your main targets. Once you've done enough damange to the fell best, he collaspes and your job is done. After that is a cutscene featuring Eowyn defeating the Witch King, and that's it for her role in the game.