When people say you take after a parent, they might mean it in looks as well as character. When they say you have their eyes, though, it might suggest something more. The eyes are the only organ directly connected to the brain and are said to be the window to the soul; they convey many emotions and thus could imply a person's deeper nature.
James Potter was a pure-blood, a Quiddich star and a reckless and sometimes arrogant student. Lily Evans, however, was a muggle-born, kind and a very good student according to Horace Slughorn. So when people keep saying Harry looks like his father but has his mother's eyes, I think they mean Harry's general behavior and his deeper nature. To the outside observer, Harry is a Quiddich star and an OK student who's sometimes prone to recklessness and anger. But deep inside, Harry is also humble, resourceful and good-hearted, like his mother was as a student.
From The Deathly Hallows:
“—mediocre, arrogant as his father, a determined rule-breaker, delighted to find himself famous, attention-seeking and impertinent—”
“You see what you expect to see, Severus,” said Dumbledore, without raising his eyes from a copy of Transfiguration Today. “Other teachers report that the boy is modest, likeable, and reasonably talented. Personally, I find him an engaging child.” [...]
“He is his father over again—”
“In looks, perhaps, but his deepest nature is much more like his mother’s."
And this might be the key to understanding his relationship with Snape, and Snape's loyalty which is eventually one of the more important elements of the series. Snape mostly despises Harry who looks so much like his father, the man who bullied Snape and then snatched the woman he loved from him. But Snape agrees to keep Harry alive no matter what because Harry is the last bit he has left of Lily. His eyes also remind Snape, a former Death Eater, that his greatest love was a human ‘mudblood’. This is why he wants Harry to look at him before he dies, to remind him of the soul he did it all for.