Saera Targaryen
This narrows it down to one character very likely to be Hugh's mother: Saera Targaryen. Baelon and Alyssa were siblings – the fourth and fifth child of King Jaehaerys I and Queen Alyssane Targaryen. Jaehaerys and Alyssane hade 13 children in total, though the only girl who was older than Baelon and Alyssa died without having children. After Alyssa, they had five sisters – Maegelle, Daella, Saera, Viserra and Gael. Saera was the most rebellious of her generation, and she really did become a sex worker later in life.
Saera's Story
In Martin's fictional history book Fire & Blood, we learn that the hot-headed Saera caused a major scandal when she was a teenager by sleeping with several men out of wedlock. When confronted, she was defiant, arguing that it was her right to do as she pleased because she had the blood of the dragon and the authority of the royal family. When her behavior was discovered, she was held under guard while her parents tried to decided how to handle this publicity nightmare. While they were deliberating, Saera escaped from her chamber and snuck to the Dragonpit, intending to claim a dragon bond and prove she was a great Targaryen. Instead, she was caught and returned to her guards.
In the end, the king and queen sent Saera to Oldtown to join the ranks of the Faith and become a Septa, but she had other plans. After a year and a half, Saera fled the Sept and the Seven Kingdoms altogether, somehow getting onto a ship and sailing across the Narrow Sea to the city-state of Lys. Lys was originally founded by the Targaryens' ancestors, the Freehold of Valyria, and it remembered many old customs and revered people with Valyrian features like Saera. Lys was also known for its "pleasure gardens" and its involvement in the sex trade – from humble sex workers to powerful courtesans. Saera found a place there and lived out the rest of her days without reconciling with her family. She eventually moved to another formely-Valyrian city – Volantis.
Hugh's Backstory
It seems most likely that Hugh is the daughter of Saera – especially since he has the typical silver-blonde hair of House Targaryen and Valyria. The only alternative is that Hugh's mother was a bastard half-sister of Baelon and Alyssa not mentioned in the histories, but that raises questions about Targaryen gene expression that neither Martin nor HBO seem interested in pursuing too far.
Still, this leaves a lot of questions about Hugh's history and upbringing. As mentioned above, Saera was a sex worker but not desperate or impoverished. She was highly respected and sought after, and was described as incredibly rich and influential. That makes it unclear why her son would find himself back in King's Landing as a blacksmith – and so poor that he couldn't afford treatment for his daughter. Hugh may fill in these gaps himself, but one way or another it seems that Saera didn't fair as well in the TV continuity as in the books.
Saera's Offspring
In Martin's book, Saera had three sons who came back to Westeros to make a claim for the Iron Throne in the Great Council, where the lords of the realm ultimately chose Viserys as their heir, passing over Rhaenys (Eve Best). These sons were treated peacefully, so there's no reason to assume Hugh wouldn't receive the same courtesy. Perhaps he'll share his own tale in future episodes, but for now fans are scratching their heads.
The House of the Dragon Season 2 finale premieres on Sunday, Aug. 4 at 9 p.m. ET on HBO and Max. Martin's books are available now in print, digital and audiobook formats.