I've described her this way before because it's pretty fun, Lucina is a time travelling warrior princess. That's a pretty good start. She has some bad*** moments, like killing the assassin going after Chrom, that prove she's not messing around. She also has moments of forced perseverance. Lucina was thrusted into a position of being a beacon of hope and strength for her people. In the scene where she's battling risen, and she saves the soldier with the line "I believe the one you want is me" and she comes face to face with Grima. Even though she's near paralyzed with fear, she doesn't back down or run away, she still does her best to point her sword at the dragon, so she can always be that beacon of hope. Then you learn more about her and you come to understand all she wanted was to be with her family, who she cares immensely deeply for.
Of course Lucina sees Marth as a role model, he's a legendary hero in her time, without any flaws. Marth did then, what she was trying to do now. He lost his parents, but still managed to take the fight to Medeus. Learning she doesn't have to rely on her persona as "Marth" in order to inspire her people is part of her character development, and I enjoy it a lot.
I'm sure I could explain better if I took a bit of time to really dig into her character, but for now that's a summary of why I like Lucina so much. I hope you can see it from my perspective!