Perhaps they are waiting for the right moment to strike. They could be hiding near a place that is important to the Order of the Phoenix or Hogwarts. By staying hidden, they can ambush their enemies when they least expect it. It could also be that they are injured and need to hide while they recover their strength to rejoin the fight on Voldemort's side.
In a fanfiction scenario, perhaps the Death Eater has a change of heart when he sees Hermione hiding. He could be secretly against Voldemort's ways and decides to help her. They might form an unlikely alliance, with the Death Eater protecting Hermione from other dark wizards. Or, the Death Eater could be under a spell or curse that makes him act differently when he encounters Hermione, leading to a very strange and unexpected encounter.
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.
She might be very cautious at first. Hermione is a smart and brave character. She would probably try to gather as much information as she could about the Death Eater without being detected. Maybe she'd use a spell to keep herself hidden while observing the Death Eater's actions.
One reason could be that the Death Eater has a family member who was saved by Hermione or her friends. So, out of gratitude, he decides to help her.
Good ones often have strong characterizations. Hermione should remain true to her intelligent and brave self, while the death eater should have a well - defined personality, perhaps with some redeeming qualities. For example, in some stories, the death eater might be conflicted about his dark allegiance when he meets Hermione.
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.
There's also 'The Unlikely Bond'. In this fanfiction, the relationship between the death eater and Hermione is explored in a very detailed way. The author does a wonderful job of creating tension and drama as they interact. It's a must - read for fans of this kind of story.
Bellatrix Lestrange's younger brother or some other minor Death Eater character could be used. This would allow for the creation of a new character with his own motives and backstory. He could be betrothed to Hermione as part of a larger plot by Voldemort to gain control over Hermione's knowledge and magic. This would put Hermione in a very difficult position as she would be tied to a very dark and dangerous family.
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 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.