Hermione would use her vast knowledge of magic. For example, she could use concealment charms to hide the Death Eater from other Death Eaters who might be looking for him. Also, she might use polyjuice potion to disguise him if they need to move around in public.
Hermione would need to be extremely resourceful. She could research ancient spells and potions that could help in the defection process. Maybe she discovers a spell that can erase the Dark Mark from the Death Eater's arm without causing him harm. She would also need to be a good psychologist in a way, as she has to convince the Death Eater that a new life away from the Death Eaters is possible and better. She would have to deal with the Death Eater's fears and doubts, and at the same time, keep him safe from any potential threats.
One possible plot could be that Hermione discovers a Death Eater who has doubts about Voldemort's cause. Maybe the Death Eater has a family member who was unjustly punished by Voldemort. Hermione, being the intelligent and empathetic witch she is, offers the Death Eater a way out. She could provide a safe haven or help them fake their death and start a new life away from the Dark Lord's influence.
One possible fanfiction could be where the Death Eater is secretly an undercover agent for the Order. He helps Hermione by providing crucial information about Voldemort's plans. This could add a whole new layer of intrigue to the story as Hermione has to decide whether to trust this unexpected ally.
It could also be that Hermione is on a mission to gather information. By helping the Death Eater, she hopes to gain insider knowledge about the Death Eater's group or their plans. This would be a strategic move on her part, using the situation to her advantage in the fight against the dark forces.
Over time, if she couldn't immediately escape, Hermione would try to influence the Death Eater. She would use her knowledge and strong moral compass to show him the error of his ways. She might start by having debates with him about the Dark Arts and the ideology of the Death Eaters, hoping to make him see the light.
She could also try to reason with the Death Eater. Hermione is known for her intelligence and ability to think on her feet. She might try to appeal to any sense of humanity the Death Eater might have, reminding him of the consequences of following Voldemort's evil path.
Well, the Death Eater might save Hermione from a dangerous situation. For example, if she's captured by other dark wizards, the Death Eater could use his influence within the dark side to free her. This would be a huge shock to Hermione and the other characters, and it would lead to a lot of mistrust initially.
Another challenge is the internal struggle within the Death Eater. His love for Hermione may conflict with his loyalty to Voldemort or the dark ideology he has been taught. He may have to choose between the woman he loves and the power and security that being a Death Eater provides. This internal conflict often makes for a very interesting and complex story in fanfiction.
In some fanfictions, Hermione as a Death Eater is a very interesting and often controversial portrayal. Sometimes, it might show her being forced into the role due to some dark magic or blackmail. This completely flips her good - girl image from the original series and allows for a very different exploration of her character.
She might react with caution at first. Given that Death Eaters are known for their cruelty, Hermione would be smart to be on her guard. She would likely start by gathering information about the escaped Death Eater, like his known abilities and any possible weaknesses.
In this type of fanfiction, the relationship between Hermione and the Death Eater could be very complex. Maybe it starts as a hostile one, but as the story progresses, they might find some common ground or mutual understanding. It could also be a story of redemption for the Death Eater through Hermione's influence.