If you're looking for a character (or group of characters) that undergo a change during The Eumenides—or even, hey, the whole of Western literature—look no further than the Furies.
At the beginning of the play, these are a bunch of thoroughly nasty ladies. No really: they're disgusting:
(Prophetess): "In front of this man an amazing band is asleep, of women, sitting on the chairs—no, I do not mean women, but Gorgons […] these however have no wings to be seen; and they are black, utterly revolting in their manner, snoring out a breath which is unapproachable […] (46-59)
These charming creatures chase Orestes over the face of the whole earth to punish him for murdering Clytemnestra. They don't do this because they have it in for him, necessarily. Instead, the Furies are just doing their job. Thus, when Apollo confronts them about it, they have no other excuse to fall back on other than that that's the way things have always been.
Also, the Furies seem a bit insecure—they are worried that, if they can't hunt down and kill evildoers, they will lose respect among men and gods. They are also afraid that the younger gods, like Apollo and Athena, are trying to take away the power of the older gods, like themselves.
In fact, the Furies are so insecure about all this that, even though they agreed to the trial of Orestes (actually, they suggested it), they threaten to lay waste the entire land of Athens after Orestes gets acquitted:
(Chorus of Furies): "You younger gods! The ancient laws—
you have ridden them down! You have taken them out of my hands for yourselves!
I am dishonoured, wretch that I am; my heavy rancour releases on this land—woe to it!—a poison, a poison from my heart to requite my grief,
dripping from below the earth, intolerable. From this
a canker destroying leaves, destroying offspring—O Justice [Justice]!—
will sweep over and strike the land
with a blight killing men." (779-787)
It takes some major finesse on the part of Athena (including some subtle threats), but, miraculously, the Furies change their tune by the end of the play. Once it becomes clear that they will still be honored as local divinities in Athens, in charge of rewarding doers of good deeds as well as punishing doers of evil deeds, they decide to give up their threats.
(Chorus of Furies): "I pray too that faction, insatiable for harm,
never clamours for this city,
nor the dust drinks its people's black blood
from counter-killings in rage,
the city's ruin its eager pursuit;
may they reciprocate joys,
resolved on sharing friendship,
and show hate with a single mind:
for this remedies much among men." (976-987)
Dang. They go from talking about "destroying offspring" and a "blight killing men" to speaking about how awesome "sharing friendship" is. How's that for a 180-degree turn? In fact, they even change their name, from the Furies ("Erinyes" in Greek) to the "Kindly Ones" ("Eumenides" in Greek). And hey, if the Furies hadn't changed their tune, and their name, we wouldn't even have a play.
More metaphorically speaking (if you think about the Furies as just a representation of revenge, instead of literally, as goddesses), if the cycle of bloody revenge hadn't ended, we might not even have modern society. So, all told, the development of the Furies as characters starts to look pretty darn important.