To understand Athena's role in The Eumenides you just need to understand a couple of basic things. First of all, Athena was the goddess of wisdom and of war. She was especially interested in protecting cities, especially her favorite city of all, Athens. (It's even named after her!)
Oh yeah, and Athena has a really weird origin story. Unlike most people, or, well, pretty much everyone, Athena was not born from a woman's womb. Instead, her father, the god Zeus one day happened to complain of a headache. To help him out, his son Hephaestus kindly split his head open with an axe. (The practice of medicine clearly gets a lot more experimental when you're dealing with immortals.) After Zeus's head was split open, who should spring out but… Athena, full-grown, and dressed head to toe in a suit of armor.
The fact that she was born from Zeus's head explains her special connection with wisdom, and especially with the wisdom of Zeus. But the not being born from a woman part also is connected with the fact that Athena wears armor, which is not typical feminine behavior. From a Greek perspective, Athena's lack of femininity is also seen in the fact that she remains a virgin for life, and never fulfills what was considered women's most important role: being a wife and mother.
Athena's actions in the play are pretty closely to what you'd expect, given this weirdo background. When Athena makes her first appearance, it is significant that she is coming from a scene of battle, where she was receiving the spoils of war from Greek military leaders. This is in keeping with her military background.
Then, when Athena doesn't jump to any conclusions about who Orestes and the Furies are and what their business is, she shows her wisdom. Her wisdom is also shown in her decision to set up a jury of mortal Athenian citizens—that way she gets to shift some of the responsibility for judging Orestes's case. Otherwise, she's bound to get a god mad at her, and that's always good to avoid, even if you are a god.
When Athena actually makes her final decision, however, it seems more based on prejudice than an open-minded judgment of the facts. That's because, even before the votes are counted, Athena announces that she intends to cast her vote for Orestes because:
"[…] there is no mother who gave me birth, and I approve the masculine in everything—except for union with it—with all my heart; and I am very much my father's." (736-738)
This looks like blatant sexism and prejudice to us. From these remarks, it seems pretty clear that Athena wouldn't approve of the Supreme Court Justice Sonia Sotomayor's "hope that a wise Latina woman with the richness of her experiences would more often than not reach a better conclusion than a white male who hasn't lived that life." Of course, Athena wasn't trying to determine the make-up of the US Supreme Court.
What was Athena doing, though? Was she just narrowly interested in the issue at hand, of whether or not Orestes was guilty? Or is it possible that what Aeschylus's Athena really wanted to do was to set up the Court of the Areopagus and thus put an end to the cycle of revenge-killings that had dominated Greek society up to that point?
If so, she might have just been looking for any old excuse to find Orestes innocent, just to get the court rolling. Does the text of Aeschylus's play allow us to come to any conclusion on this question? In any case, it's clear that, once the trial is over, Athena shows herself a master of social graces. Through a combination of promises and threats, she convinces the Furies to abandon their old ways and become "Kindly Ones" ("Eumenides") in charge of maintaining a positive social atmosphere in the city of Athens.