While the name is greatly significant to the story (remember the carving from 1500 that Lockwood notices over the door at Wuthering Heights?), this son of Hindley and Frances is born with significant disadvantages. Mom died when he was a baby, and Dad was too busy drinking and being abusive to care. Poor Hareton becomes a victim of everyone else's need for revenge.
Heathcliff treats Hareton just like his Hindley treated Heathcliff – like a laboring, uneducated oaf not deserving of any family privileges. And Hareton is not helped by his resemblance to Catherine. His eyes are an uncanny match to his aunt's, and he looks far more like Catherine than her own daughter does.
Of course, he does become the beneficiary of the novel's happy ending: he gets the girl and the house, and he learns how to read. Hareton manages to transcend his brutal mistreatment and evolves from an illiterate brute into a kind and compassionate friend (and eventually lover) to Cathy Heathcliff. In a sense, Hareton redeems the Earnshaw family by breaking the pattern of abuse with which he was raised, earning back the property, and just being an all-around decent guy. Cathy Heathcliff falls in love with him because she senses that underneath his rough exterior, Hareton feels sympathy. The nature of their love is quite different from Catherine and Heathcliff's. It is characterized not by drama and abuse, but by kindness and serenity.