This practice of using eyes comes about because eyes are the windows to the soul. They express what the author wants you to know about their characters. Also, it's a quick fix if you just can't find the right actress to play your character.
I went to a few avatar sites that had entire sections devoted just to eyeballs, picked a few and posted them here to show you what you can expect if you see these Sues staring back at you.
She's a bad ass bitchiwitch. She has been studying Dark Arts since she was a fetus. She's angry at the world and she lives in a Slytherin-phobic society. She believes she has every right to be as hateful as she is. She thinks there is nobility in her blind hatred. The only joy she gets in life is ruining Ravenclaws, hurting Hufflepuffs, gravely injuring Gryffindors and even slaying fellow Slytherins.
She wants- no, NEEDS to be Draco Malfoy's girlfriend. She hopes one day, if she hangs around him long enough, he will see how she is his perfect mate. She's incomplete without him. She likes what he likes and hates what he hates. She is totally the yin to his yang because she was designed that way.
tshirts that say snotty things on them. She says she doesn't care what others think of her. She isn't afraid to look foolish which is good because she always does. She finds herself very amusing. She's a danger to herself and others. Don't dare her to stick a pencil in her eye. She'll do it!