Desperation. If she saw no other way to stop Voldemort and the Death Eaters' reign of terror, she might think that infiltrating them as one of their own was the only solution. So she could become a Death Eater in that sense.
Betrayal might be a factor. If she felt betrayed by the wizarding community she was trying to protect, she could be pushed towards the Death Eaters. For example, if she was wrongly accused of a serious crime and the so - called good wizards turned their backs on her.
Bellatrix Lestrange is also a likely candidate. Her encounter with Hermione would be extremely intense. Given her loyalty to Voldemort and her rather unhinged nature, she would likely pose a serious threat to Hermione, but Hermione might also find a way to use Bellatrix's madness against her.
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.
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.
There's also a possibility that it could be a minor Death Eater that we don't know much about from the original series. The author of the fanfiction could create a new character who was a Death Eater but is now having doubts. Hermione could be the one to find this person and start their journey of change. This would give the author a lot of creative freedom to develop the character's backstory and their relationship with Hermione.
It might be portrayed as a tense encounter. Hermione would be on her guard, ready to use her magic if necessary. The Death Eater, on the other hand, could be sneering and full of dark arrogance.
Hermione would likely try to outwit them. She has a sharp mind. She might use her knowledge of magic, both dark and light, to find ways to counter the Death Eaters. For example, she could use spells that they wouldn't expect or find loopholes in their dark magic.
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.
One possibility in this fanfiction could be that Hermione is forced to become a Death Eater due to a powerful dark magic spell that controls her mind. This would lead to a great internal struggle as her original values clash with the cruel and dark ways of the Death Eaters.
There's 'The Unlikely Bond'. Here, the Death Eater, say Draco (as in some fanfictions he has Death Eater - like tendencies), falls in love with Hermione when they are paired for a school project. It shows how love can cross the boundaries of their different beliefs. Also, 'Shadow of Love' is a great one. The Death Eater is secretly protecting Hermione from the other dark forces, all because of his unspoken love for her.