Character Analysis
Although a woman, Brienne of Tarth accepts the gender roles of a man in Seven Kingdoms' society. Dresses, dances, and making babies are not for her—instead, she's a knight ready to ride off and save some damsels in distress… or whatever the male equivalent of a damsel is. Dudes in distress?
This Is a Man's World
Brienne chose to enter Westeros's male sphere of influence (read: war, fighting, knighthood, and such) rather than the female sphere of influence (read: raising children and home management). More importantly, she's really—like, really—good at the jobs generally left to the Seven Kingdoms bros.
When we first meet Brienne, she's in the middle of a tournament melee put on by Renly. Among her opponents is none other than Loras Tyrell, the famed Knight of Flowers. In previous tournaments, Loras has toppled opponents as feared as the Mountain and Jaime Lannister, but Brienne manages to defeat him, proving she can rock'em and sock'em with the best of 'em.
Late in the story, she and Catelyn watch Edmure and Tywin's forces engage each other at the Red Fork. Brienne demonstrates she has a tactical mind for warfare by providing Catelyn with a play-by-play of the events and explaining what the commanders must be thinking given their units' formations (46.Catelyn.52).
She also values the traits associated with a knight's life, such as courage, honor, and chivalry, and she is open minded enough to see these traits in untraditional places. Catelyn may not be a warrior, but Brienne admires her "kind of woman's courage'" (40.Catelyn.56)—courage, for Brienne, is something more than just the ability to swing a sword, even if she can't put why into words. In fact, she admires Catelyn's form of courage enough to swear herself to her service.
Paying the Price
Brienne is sometimes called Brienne the Beauty, but it's not because she's comely. Rather, it's a way for people to mock her appearance and her choice to engage in gender roles not traditionally associated with women.
Catelyn notes Brienne's hair appears as dirty straw, her teeth are crooked, a thousand freckles decorate her face, and her nose has been broken several times. Looking at her Catelyn thinks, "Is there any creature on earth as unfortunate as an ugly woman?" (23.Catelyn.67). Also, even though she wins the melee, Brienne receives nothing but scorn from the spectators (23.Catelyn.59). Boo and hiss to them, we say.
Brienne evidently suffers as a result of the mocking she receives—Catelyn notices that she speaks only to answer someone and seldom lifts her gaze to meet another's eyes (23.Catelyn.96). Her personality is stoic and introverted, preferring exclusion from others. On the ride to Riverrun, Brienne does not chatter, cry, or laugh, and although she rides with them every day and night, she "never truly becom[es] one of them," and Catelyn decides Brienne keeps herself enclosed for protection, comparing her nature to the walls of Winterfell (40.Catelyn.27-28).
Clearly the years of social abuse have taken their toll on Brienne's sense of self worth and her ability to feel accepted by others.
Love, Duty, and Puppy Eyes for Renly
This social abuse could explain her feelings toward Renly. When Brienne wins the melee, Renly eagerly accepts her into his Rainbow Guard, and although this isn't explicitly stated, we're willing to bet it's one of the few times Brienne has been accepted eagerly into any club or organization. But it goes further back than even this acceptance. As Ser Cornay remembers, "She loved Renly Baratheon from the first moment she laid eyes on him, a blind man could see it" (43.Davos.29). She only has eyes for Renly, it seems.
We can't say why Brienne admires Renly so, but we can say she continues to. When she enters Catelyn's service, her one condition is that when the time comes she will be allowed to seek revenge against Stannis. But that time does not come by the end of A Clash of Kings, so we'll have to wait and see what happens.