He was born in 2303, and his birth unknowingly marked the beginning of the Narnian Revolution.
Up until that year, his cousin, Prince Caspian, had been safe at his home in the Telmarine-ruled Narnia. Because Miraz had had no children, and thus no heir, he had kept Caspian alive only so that someone of his family would one day inherit the Telmarine throne.
It was only after the birth of his only son that Miraz then attempted to have Caspian assassinated, which then sent him into the arms of the Narnians, igniting the war that eventually reclaimed the Narnians their homeland and Caspian his throne.
It is unknown what became of Miraz's son, after his father was killed by Lord Glozelle in the war, but it's most likely that after Caspian was made king, Prunaprismia took her son and left Narnia with the other Telmarines, or possibly to another country somewhere in the Narnian world.
AdaptationsIn the Disney adaptation of Prince Caspian, the biography of Miraz's son is pretty much the same as it was in the book.
The few exceptions are the added scenes where Miraz and Prunaprismia spend time with him near the start of the film, and the one where they present him to the kingdom, at Miraz's coronation.
There were also a couple of deleted scenes in the film; one, which was an extended version of the scene where Miraz holds him while looking out over the balcony, awaiting the capture and return of Caspian. Prunaprismia approaches him, asking if he is showing his son the kingdom, not knowing that he had secretly tried to kill Caspian.
Miraz claimed he was just getting some air, but then makes a remark that his son should get to know the kingdom, if he is to rule it one day, indicating his plans for his son to rule.
- "Showing your son the kingdom?"
"Just getting some air. But you're right, he should see it, if he is to rule it one day."
- ―Prunaprismia and Miraz [src]
There was also one deleted scene, where Peter and Susan accidentally end up in the baby's nursery, and subsequently see Miraz's son in his cot, while looking for Miraz's room. They left him alone, and went on to continue searching for Miraz.
- "Where did that come from?"
"Mum never had that talk with you?"
- ―Peter and Susan [src]
It was also confirmed in the film that Miraz's son definitely left Narnia with his mother, grandfather and General Glozelle (who had not killed Miraz in the film, and was portrayed as a more honorable man), to go live on their new home on Earth, on the deserted island in the South Sea.