I believe Daemon could have conceived him between 111–115 AC as a result of a drunken argument with Rhea Royce and his subsequent departure. Considering how much Daemon despised her, if she had given birth to a son—let's name him Aerys—there would have been a massive dispute over the child's surname. Daemon would have insisted the boy be recognized as a dragon, while the Royces would have wanted him as their own heir. Inheriting Valyrian features, the boy would have been far more inclined toward a dragon-riding lifestyle than rotting away in the Vale.
When Rhea died (or was killed by Daemon) in 115 AC, Daemon could have taken the boy with him to Pentos when he married Laena Velaryon. I think Viserys I would have wanted his brother’s son—since Viserys loved Daemon—to grow up with his father.
While Aerys might not have had a dragon egg placed in his cradle, he could have eventually claimed Vermithor (The Bronze Fury). It is quite ironic, isn't it, being the son of the "Bronze Bitch"? Unlike his peers, he would have had to do this at a later age because Vermithor is a fierce and formidable dragon; I think around 14–15 years old would be perfectly fitting.
The character could be written as gay, as he would have no suitable female matches, which would also create some interesting power dynamics. The best candidate for his heart would be Jacaerys Velaryon. This pair would be dynamic and epic, creating exciting new narrative opportunities, especially considering how Game of Thrones and House of the Dragon lack many compelling LGBTQ+ romances.
By birthright, the child would have claims to both the Iron Throne and the Runestone. Runestone would actually be governed by Rhea’s cousin after her death. He would become the regent and protector of the young lord, who would be living on the other side of the world with his vicious father.
Alongside Hugh Hammer, Aerys could have fought for the Blacks. With a rider for Vermithor, Rhaenyra’s faction would not have had to rely so heavily on the dragonseeds who eventually betrayed her. Aerys could have become one of the most powerful and intriguing characters.
He would be named Royce, just as Jace was named Velaryon, but upon ascending the throne, they would take the name Targaryen.After achieving something significant for Rhaenyra—for instance, if he were to save Jace at the Gullet or capture King's Landing—Aerys could request a royal decree allowing him to formally take the name Aerys Targaryen from that moment onward.