Greetings! I have only two things to add to the previous answers. Well, make that three! First, that the myth is an example of the "Rash Promise" theme, that is making a promise before you know exactly what is being asked. The way to enforce that a god would in fact fulfill a promise was to make the god swear by the River Styx to do so. The cost of not honoring the promise would then be the god's immortality!
Second, it may not be clear in Hamilton, but the myth is rooted in sexual shame. The boy has been raised by his mother without a father present. His mother has told him that his father is Helios, god of the sun (in some accounts this is made Apollo but for logistical reasons that makes no sense: Apollo has other things to do during the day!) but the boy has his doubts. He is teased by the other boys. He is in their minds a bastard--a boy whose father didn't marry his mother. Worse, someone tells him that whoever seduced his mother was only pretending to be a god. Many a young man has charmed his way into a woman's bed by claiming to be a god, they tell him! So the boy NEEDS to prove to himself and others that he is the son of a god. This is revealed in Ovid's telling of the tale in the conversation between the boy and his father. The boy says things like Father, if you are my father... and the god answers Son, for you are surely my son....
Finally, the third point is that although Hamilton is an excellent summation of many myths, it is very useful and often more enjoyable to read the myths in original sources where there is more detail and often more racy details to be found which can help clarify meaning!
Phaeton asked his father, the Sun God, if he could control the sun chariot.
When Phaeton sought out him out, the Sun God said that he would grant his son any wish his heart desired. With that, Phaeton asked his father the unthinkable: "I choose to take your place, Father. That is the only thing I want. Just for a day, a single day, let me have your car to drive." Phaeton never relented in his wish. Even though his father repeatedly warned him that he could not do it, Phaeton had his heart set on it. The sun god dreaded the proposition, even telling his son that his fear was proof that he was the boy's father.
Despite his father's warnings and fear, Phaeton got his wish. Predictably, the results were bad. Phaeton could not control his father's chariot. He was ill-equipped to control it. His story ends when he falls from the car and dies. The ending to the story connects his great spirit of hope and daring with an immense failure.
The myth of Phaeton demonstrates how wishes might be more than what they appear to be. Phaeton's wish, something that he yearned for more than life itself, turned out to be worse than he could have ever imagined.
Phaethon was the son of Helios , who was the god of the sun and Clymene.When Phaethon reached to certain age he learned that his father is the god of the sun.After Phaethon discovered about his father so he decided to meet his father and he went on a journey to the east where he found his father Helios.One day he asked his father for a favor that he would drive the sun chariot so that he can prove that he is the sun of the god Helios.He couldn't control the sun chariot so the earth was in danger of burning so Zeus killed Phaethon by thunderbolt to prevent further disasters.