In many 'Game of Thrones' fanfics, Daenerys Targaryen is often seen as the'mother of dragons'. But if we consider the idea of a 'father of dragons', some might point to Aegon Targaryen from the past in the lore. He was a powerful Targaryen and dragons were a big part of the Targaryen dynasty which he was a part of.
Another possibility could be an ancestor of the Targaryens who first discovered or harnessed the power to create or control dragons. Since the Targaryens are so closely associated with dragons in the 'Game of Thrones' universe, it's likely that the 'father of dragons' in fanfic would be related to their family history in some way.
One common plotline is that the 'father of dragons' is trying to bring back the old glory of the dragon - riding Targaryens. He might be on a quest to find more dragon eggs and hatch them to build an army of dragons and reclaim the Iron Throne for his family.
In many 'Game of Thrones' fanfics, different characters could be considered the 'real' lady of Winterfell. It could be Sansa Stark, as she has a strong claim to Winterfell through her Stark heritage. She has endured a great deal and learned the ways of politics, which would make her a worthy lady of the keep.
The dragons in 'Game of Thrones' are truly magnificent creatures. Daenerys Targaryen hatches the dragon eggs and raises them. They are a symbol of her power. Drogon, in particular, is very important. He is the largest and fiercest of the three dragons. He helps Daenerys in her conquests, like burning down the enemy's ships and fortresses.
Another possibility is that the father could be a new character introduced in the fanfiction. This new character could be a powerful lord from a far - off land who comes to Winterfell, falls in love with Sansa, and they have a relationship. Or perhaps it's a mysterious figure with magical powers who impregnates Sansa as part of a grander scheme in the magical and political landscape of Westeros.
Daenerys Targaryen would be an obvious choice. Since she was the 'Mother of Dragons' in the original series, her return or a continuation of her story in a fanfiction where the dragons come back would be very fitting. She could have new adventures and use the dragons to reclaim her rightful place on the Iron Throne or to bring justice to Westeros.
She could interact with Daenerys Targaryen. They are both strong female characters with special powers. They might initially be rivals, both vying for power and influence in Westeros, but then realize they can achieve more by working together. For instance, Elsa could use her ice powers to cool down the hot deserts that Daenerys has to cross with her armies.
Perhaps the main character could be a female machinist. In a world dominated by men, she has to fight against prejudice while also using her skills. She could be from one of the noble houses but has a secret love for machinery. She might be working on a project to build a flying machine, which is seen as crazy by others, but she is determined to prove them wrong and change the face of warfare in Westeros.
A lost princess from Westeros who washes ashore in Ulthos. She quickly realizes that Ulthos is a land full of secrets and danger. With her charm and the political knowledge she learned in Westeros, she tries to navigate the complex power structures in Ulthos and make a name for herself.
A demon who has taken on human form could be the main character. He could be disguised as a knight or a noble, secretly using his demonic powers to influence the events in the Seven Kingdoms.
A samurai - like character from Zipang could be a main character. They would be skilled in combat and bound by a strict code of honor. They could enter the fray in Westeros and become involved in the power struggles, perhaps even challenging the likes of Jaime Lannister in a duel of honor.