She would likely be on high alert. Hermione is a smart witch, so she would quickly assess the situation. She might immediately reach for her wand and prepare to defend herself. Her first reaction would probably be fear, but she would soon overcome it with her determination to not be a victim.
Hermione could also be very defiant. She's not one to back down easily. So, instead of just running away or cowering, she might stand her ground and try to engage Rodolphus in conversation. This could be a risky move, but she might think she can gather some information or at least buy some time. She could ask him what he wants or why he's there, all the while ready to use her magic if he makes a wrong move.
Well, one possibility is that Hermione would be in immediate danger. Rodolphus Lestrange is a Death Eater and quite a dangerous one. He might try to capture Hermione to use her as a bargaining chip against the Order of the Phoenix or just to cause harm. Another scenario could be that Hermione tries to outwit him using her intelligence and knowledge of magic. She could try to trap him in a spell or find a way to escape his clutches. Maybe she would also try to gather information from him if she thought she could manage it without getting caught.
It could start with animosity. Rodolphus has been a Death Eater and Hermione is a Muggle - born, so there would be initial hatred. But if they were forced to work together, say in a dangerous mission to save the wizarding world, they might slowly gain respect for each other. Hermione's knowledge and Rodolphus' experience could be complementary.
She could also be calculating. Since she's so intelligent, she might be quickly assessing her surroundings, looking for any weaknesses in their captivity of her. She might be thinking about which spells she could use effectively in the given situation, or how to use the environment to her advantage.
Harry Potter is a very likely character to come to Hermione's rescue. He and Hermione are best friends, and he has a strong sense of loyalty. He would surely do whatever it takes to save her. Ron Weasley is also a candidate. He cares deeply about Hermione and would be extremely worried. He would join the rescue mission without hesitation.
One plot idea could be that in an alternate universe, Rodolphus, after realizing the error of his ways in the wizarding war, tries to make amends by protecting Hermione from a new dark threat. This could lead to a begrudging partnership that slowly turns into an unexpected friendship or more.
In some fanfictions, there could be a situation where Hermione forms an unexpected bond with Rodolphus. Maybe she starts to understand his backstory and what led him to become a Death Eater. This could lead to a complex relationship where she tries to redeem him in a way. Or perhaps, Rodolphus discovers that Hermione is far more powerful and resourceful than he initially thought, and this throws his plans into chaos as he has to constantly adapt to her actions.
Hermione is known for her strong sense of justice and intelligence. The Lestrange brothers, coming from a dark pure - blood family, might have a more cavalier attitude towards the law and a different view on power. Hermione would clash with them when it comes to ethical use of magic. For example, she would never condone their use of dark spells for personal gain. Her studious nature would also contrast with their more impulsive and power - hungry ways.
Hermione would probably be deeply hurt at first. She trusts Harry and Ron so much, and being abandoned by them would be a huge blow. But she's also very determined. So, she might throw herself into her studies even more, using her knowledge as a way to cope and also to find a solution to the situation.
This would completely transform her story. Hermione's academic achievements might be seen in a different light, perhaps as an attempt to distance herself from her family's dark past. Her interactions with the Order of the Phoenix and other anti - Voldemort groups could also be altered. She might face more suspicion initially but also have unique insights into the Death Eaters' minds due to her family ties.
If the future she visits is a bad one, Hermione would be determined to change it. She has a strong sense of justice and wouldn't stand by and let people suffer. She would use her knowledge of the past and her wits to find a way to make things better, whether it involves changing the actions of certain individuals or influencing the course of events on a larger scale.