Harry- He was actually mostly average, on average, and I have long been weary of people citing Harry as a successful student (though I will admit that you didn't refer to him as a successful student, just “successful at Hogwarts”). It's unrealistic to believe that he could have become an Auror given the scholastic requirements, but it's possible that there are setups in place that allow a sufficiently high score on the practical exam (or a sufficiently biased proctor or grade-marker) to surpass this. On the other hand, he really did have natural talent in a couple of his classes, and did sometimes (often?) stoop to cheating off of Hermione or a marked-up Potions textbook in other classes.

Hermione- She got all of her schoolbooks nearly a year before the start of term (most pureblood families probably waited until the end of summer before first year), and had immersed herself in the culture of learning long before she found out she was a witch. While she couldn't use a wand at home, pureblood children aren't allowed to do magic at home either. Some families (such as the Longbottoms, and Molly but not Arthur Weasley) seem to actually respect that rule, and theory is shown to be just as important as practice in many parts of magical study (more important in purely rote classes such as the entirety of History of Magic). The “culture of learning” thing mentioned above is also a bigger advantage than it would be in most of Muggles' so-called first-world civilization, considering the vast majority of wizards tend to take the spoon-fed lessons and low-hanging fruit of knowledge as everything they need without doing much in the way of critical thinking of individual study (outside of the near-completely ignored Ravenclaw house, which is suspiciously lacking). In addition to that, she worked hard to stay ahead of the curve. She happened to be powerful for her age, but not outside of the implied normal deviation (that being, most of her skill came from memorization and understanding of theory).

Lily- I'm chalking that one up to author favouritism. There's no real reason for her to be a prodigy, but she is one, and “author favouritism” fits the pattern of how Rowling tends to treat her as a character.

Tom Riddle- It's a combination of a little of the same and a lot of the opposite compared to Hermione. He spent his summers in Hogwarts when all students with family went home, where there was nothing to do but explore or study. That's the “same” part- he had more reason and time to study compared to the other students. The “opposite” part is that Tom Riddle was an incredibly powerful wizard, and had been practicing wandless magic for what could have been years before he was eleven, focusing what the administration would falsely call “accidental magic” in ways that would not be possible for even many seventh-year students.