AFFC Spoilers Ahead...
Well, Brienne's POV does not further the "Game of Thrones" plot, the "War against The Other", nor the "What's up with Eddard's children?" plot. Brienne is trying to find Sansa and Arya, too... but we already know what's up with the girls, so upon first glance Brienne's story may appear to be pointless. I think all of us felt this way upon first reading Brienne's chapters, but I think her chapters clue us in to a few important aspects of the overall story.
First, Arya's POV in the first three books was the only one that really showed the lives of the common folk. Sure, we all like stories about dragons and kings, but Arya gave us glimpses of the effect of the Game of Thrones upon the populace. Jon's POV is fairly gritty. It deals with the common men on the Wall and the Free Folk, but the North is far removed from the Game of Thrones. Brienne's POV fills this gap. Her travels reveal the social unrest from Lord Tarly's occupation, the real economic struggles the small folk have, the fear of rampant lawlessness, and the religious movement of the masses trying to give some meaning to the social/cultural/economic upheavals.
Second, I think that atonement (or at least self-impovement), service, and self-sacrifice in the name of justice are major themes of ASOIAF. Eddard is the prime example of sacrifice... He sacrificed himself for Lyanna, for Jon, for Robert, and finally for his daughters. But mayhaps we've read Eddard wrong, mayhaps he is trying to atone for something...
Jaime is the best example of atonement in the story. His crimes are many; regicide, attempted infanticide, adultery, high treason (against two separate dynasties), murder, aiding and abbetting high treason, etc... I think he has the desire to repent, but will he go the distance? I could not say.
Davos demonstrates GRRM's theme regarding service. Eddard does, too, but Davos gives a fuller picture. He serves in whatever capacity is asked of him. He serves with Stannis' best interests at heart and not his own.
Brienne embodies all three of these themes. Atonement... She may not have commited egregious sins, but Jaime lost his hand on her watch and she has yet to fulfill her promise to return Catelyn's daughters. Plus, she's vitally important to Jaime's redemption plot.
Service... She is a sword sword of Catelyn Stark. She longs to prove herself and gain the social status and economic safety of knighthood, even if she never gains the title. Podrick Payne, as her squire, gives Brienne the opportunity to be in command of someone serving her... I think that's a nice twist.
Self-sacrifice... Brienne has not returned to the comfort and peace of her father's halls. She's fought and killed to fulfill a promise to a dead woman. She's been captured and hung.
Third, the theme of worth... the possibility of meritocracy... or at least everyone being "created equal" and "endowed by their Creator with certain unalienable rights"... comes shining through in Brienne's POV. Jaime, Cersei, Joffrey, Robert, Oberyn, Arianne, Theon, Viserys, and others are all good looking, if not down right paragons of beauty... yet they are all at least walking disasters, if not the devil incarnate. On the other hand, Tyrion, Brienne, Arya, and Sam (the Hound may get included on this list later, but right now he's a bad, bad man) are nothing to look at... some of them are down right ugly, but we've gotten glimpses of their characters... they long to do justice, they long just to be themselves, they tend to do the right thing over serving their own self interests (okay, so Tyrion does both when he can). Brienne's POV is there to show that people should be judged upon their merits, not their looks... Which leads me to the last purpose for Brienne's POV.
Finally, Brienne's choice in shield decor is the most obvious and tangible link to Ser Duncan the Tall. Not even Jaime's ruminations through the White Book nor Barristan's tales provide this. Only Maester Aemon's memories of Aegon help link the past with the present in Westeros.
