What's fascinating about Athena is she embodies a traditional vengeful-goddess role, but how she carries out that vengeance is more complicated than one would expect. Her presence is felt throughout the play because she offers second chances and lessons as much as revenge.
Athena’s role in Ajax’s downfall is revealed out of order in the text, but chronologically when we first see her interact with Ajax he rejects her authority in battle, claiming he can hold the line himself. The messenger says this is the moment where he earns the hatred of the goddess. Some time after this, Ajax loses the contest for Achilles’ armor in a competition of speech-making. Wits and words are Athena’s specialty, so naturally her favorite, Odysseus, wins. This is the event that causes Ajax’s breakdown, so I believe in this Athena's revenge is complete. Interestingly, Sophocles puts that whole arc, which could have been a complex play in and of itself, offstage in the mouths of minor characters and focuses on what happens because of it.
This brings us to Athena’s only scene onstage, at the beginning of the play. She’s just protected Odysseus and the generals from Ajax’s wrath and caused him to slaughter farm animals instead, and then she makes Odysseus look on Ajax in this state of madness. I found that an interesting gesture and wondered why she considered that so important when he clearly didn’t want to see him. She says it’s a reminder to stay humble, that even the strongest can be brought low by the gods, but we get another answer at the end of the play. When Odysseus reappears and defends the men who want to bury Ajax, he says, "I hated him [Ajax] when it was honorable to hate him." At first reading it seemed like he just meant not to speak ill of the dead, but the moment it stopped being honorable to hate Ajax was when he saw what he'd become. It's one thing to hate a man who tried to kill you. It's another when you realize he's not in his right mind and acting out of a place of total brokenness. That's why it was so important to Athena for Odysseus to see him. That's one major way her presence is felt; her actions cause Odysseus's change of heart and bring the play to its conclusion.
She also continues interacting indirectly with Ajax as he comes to his end, and I believe it’s less vindictive than the characters understand it to be. Ajax, of course, sees her tricking him as a torment and further revenge, because she kept him from his goal. Ultimately, though, if he had slaughtered and tortured Odysseus and the generals as he planned, that would have been a much darker fall. In this way she spared Ajax. Killing farm animals is horrible, but it’s perhaps easier to come back from than killing comrades-in-arms. Later a messenger arrives, bearing a message from Calchas the seer that if Ajax stays in his tent for one day, he may yet be saved. “The savage Fury of goddess Athena will hound him for one day, and no more.” This, again, seems kinder than what we know of the Greek gods’ vengeance. Poseidon’s grudge takes Odysseus the entirety of the Odyssey to escape, but apparently Athena puts a 24-hour limit on hers. She’s clearly powerful enough to continue hounding him, so this isn’t a question of her ability. What, then are we to make of her not only making it so easy for Ajax to escape her wrath but telling everyone how it can be done? That seems, again, like an offer of an armistice. If these, then, are second chances instead of vengeance, Athena is a more complex character than the usual vengeful deity. She punishes Ajax for his arrogance, but she’s not needlessly vindictive. The tragedy is that despite that, the first damage is still done and Ajax still dies.
I found it interesting that Ajax and the duo of Athena and Odysseus represent the two sides of war. Ajax represents the passionate, powerful, raging side of war, the side that feels strongly the righteousness of one’s cause. Athena, on the other hand, is the strategy, the cold, calculating side of war, which partly explains their disconnect. In Brian Doerries’ Theater of War, he tells a story about Major Hall, who suffered a similar disconnect from his commanding officers in Iraq. Major Hall believed “if you want to go home” from a war, “you will be the aggressor.” The strategies of his generals, which more often than not put his men in danger and were out of touch with what was going on at the front, left him disillusioned and depressed, much like Ajax. The generals clearly think they’re doing what’s best for the war effort, just as Athena tries to spare Ajax again and again, but when the strategists aren't working together with the warriors on the front lines, problems arise every time.
Athena’s presence in Ajax is unlike that of any other character in the play. Though she’s only onstage for one scene, her intervention both heightens the tragedy through Ajax’s missed opportunities for redemption and brings it to a conclusion through Odysseus’s learned compassion for his enemy.