I was first introduced to Lucina when I came across an unfamiliar artwork of Lyn wearing gear I had never seen her wearing when trying to find decent artwork images of her so that I could revamp her shrine. My initial thought was that it was fanart, because it was a totally different looking Lyn, but then I discovered that it was actual in-game art for her DLC character in Fire Emblem: Awakening. From there I was introduced to the game and the excitement ensued. When the game finally came out in North America, and I was able to use Lucina, my liking of her was amplified, if not solidified. She was absolutely amazing.
It's hard to put into words why she's one of my favorite female character, but I will try nonetheless. A common factor that makes a character appeal to me is when they try to overcome some sort of obstacle. While Lucina doesn't quite work at it the same way as some of my other favorite characters she does still work at it, and tries the best she can while she's at it. Although her intentions to change the past so her future doesn't come to pass may seem selfish it really isn't. She doesn't do it for herself, but instead she does it for OTHERS so that they don't have to experience her dark future. Although some things she tries to prevent seem to still find a way to happen (ie: Emmeryn's death) she still tries her best and she doesn't give up.
The way Lucina acts is also a contributing factor. She doesn't flaunt her station, and instead works toward helping her friends and comrades in accomplishing their goals and dreams. She also isn't afraid of getting her hands dirty and tries her best at anything and everything she does and helps in any way she can, expecting nothing in return for it. It's such a rare thing to see in today's world so I greatly admire anyone who can do that. Although I try and do it as well my efforts still pale in comparison to hers. Still, just simple things like holding a door open can make all the difference.