It could be that the Death Eater is under a powerful spell or curse that forces him to help Hermione. This spell could have been cast by an unknown wizard or witch who wants to use the Death Eater's skills and knowledge to help Hermione in her fight against the dark side. Another possibility is that the Death Eater has a personal vendetta against another Death Eater or a dark force that Hermione is also fighting against, so they form an unlikely alliance.
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.
The development might also involve the Death Eater introducing Hermione to some secrets of the Death Eater organization in a bid to undermine it from within. This would be a very risky move. Hermione, being the clever witch she is, would have to be very cautious in dealing with this new - found ally. She would need to constantly assess whether his help is genuine or part of a bigger trap. Their relationship would be full of tension and mutual suspicion at the start, but gradually they could learn to trust each other more as they work towards a common goal, like bringing down Voldemort's regime.
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.
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.
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.
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 '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.
One possible aspect of this fanfiction could be Hermione's encounter with an unexpected Death Eater. Maybe she stumbles upon one during a mission or while exploring a hidden area. This encounter could lead to a series of thrilling events, like a battle of wits or a dangerous duel.
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.
Another idea could be that she is somehow brainwashed. There could be a dark spell or a powerful magical object that alters her perception and makes her believe in the Death Eater ideology. In this scenario, her friends would have to find a way to break the spell and bring her back to her true self.