She might use her intelligence to quickly understand the complex nature of time travel. For example, she could master the TARDIS controls faster than others. Her studious nature would help her learn about the Time Lord's history and culture, allowing her to blend in better when interacting with other Time Lords.
Well, Hermione is already very good at handling complex magical knowledge in the original story. As a Time Lord in the fanfiction, she would probably apply her quick - learning skills to the new abilities. She might struggle at first with the moral implications of time manipulation, but eventually use her strong sense of right and wrong to make good decisions regarding time travel.
In this fanfiction, Hermione's adaptability to Time Lord abilities would be quite interesting. Her magic skills could potentially be enhanced by Time Lord technology. She may be able to combine her existing spell - casting with time - based powers. Also, her friendship - building skills could help her form alliances with other Time Lords, which would be crucial for her survival and success in the Time Lord society.