We aren't actually given too many details regarding his birth. It is an event that is not witnessed by the narrator of this section, Nelly Dean, as we are told that she was working in the fields when he was born. However, the birth of this "grand bairn," as he is reported as being, is accompanied with sadness and grief, because it looks extremely likely that his mother will die because of the consumption that she has suffered from and the exertion of giving birth. Note what the girl tells Nelly about the birth and her mistress, Hindley's wife:
"The finest lad that ever breathed! But the doctor says missis must go: he says she's been in a consumption these many month. I heard him tell Mr. Hindley: and now she has nothing to keep her, and she'll be dead before winter. You must come home directly. You're to nurse it, Nelly: to feed it with sugar and milk, and take care of it day and night."
Thus Hareton entered the world with great promise concerning his health and appearance, but also with great sadness, because giving birth to him also has sapped the strength of his already weak and ailing mother. It is a sign of the happy ending of the novel that the promise with which Hareton started life is fulfilled in his union with the younger Catherine and his education.