One idea could be that Elijah tries to court Elena in a very old - fashioned and gentlemanly way. He might take her on trips to historical places he has known over the centuries, sharing his stories and knowledge with her. This would be a contrast to the other vampires' more aggressive approaches towards Elena.
Perhaps Elijah wants Elena because she reminds him of someone from his past. He could have known a girl long ago who had the same spirit as Elena. He then becomes obsessed with protecting her and winning her love, all the while struggling with his own dark vampire nature that could potentially drive her away.
In a fanfiction, Elijah could want Elena because he sees her as a key to changing the vampire - human relations. He believes that if he can win her over, she can help him bridge the gap between the two species. So he starts by protecting her from other threats in the supernatural world, gradually earning her trust.
A fanfiction could explore the idea that Elijah's desire for Elena is part of a bigger prophecy. Maybe there's a hidden message that says if Elijah and Elena are together, it will bring peace to the vampire and human world. So he's driven by this sense of duty as well as his personal feelings for her. This would lead to a lot of internal conflict for Elijah as he tries to balance his feelings with the potential consequences of their relationship.
One common antagonist could be Klaus. He has his own motives and often clashes with Elijah. His presence can create a lot of tension between Elijah and Elena as well, since he might target Elena to get to Elijah or vice versa.
One interesting storyline could be Elijah protecting Elena from other powerful vampires. Since Elijah has a sense of honor, he might see it as his duty to keep her safe.
One interesting storyline could be Elijah protecting Elena from a new and powerful enemy in Mystic Falls. Maybe an ancient vampire who has a vendetta against the doppelganger. He could use his centuries - old knowledge and skills to keep her safe while also trying to understand his growing feelings for her.
One idea could be that Klaus, after being away for a long time, returns to Mystic Falls and finds Elena in a difficult situation. He decides to help her out of a sense of an old, unspoken connection. As they spend time together, they start to understand each other better and a relationship begins to form.
One possible plot could be that it's a magical or unforeseen situation that leads to their pregnancy. Maybe a powerful witch casts a spell that binds their fates in this way. They then have to deal with the reactions of the other characters, like Damon and Klaus. Elijah, being an original vampire, would be worried about how this new situation affects his status and power, while Elena would be thinking about how it changes her life in Mystic Falls and her relationships with her friends.
In other fanfictions, Elena might be in a desperate situation, like being poisoned by a new type of vampire venom. Elijah, with his vast knowledge of vampirism, tries to save her. During the process, he discovers her strength and resilience. This event makes him see her in a new light, and Elena, in turn, is grateful to him. Their relationship then develops from a simple acquaintance to a deep emotional connection that could potentially lead to a romantic relationship.
Their relationship would likely become more complex. Elijah is an old - fashioned vampire with a sense of honor. If Elena is pregnant, he would feel a strong sense of responsibility towards her and the unborn child. Elena, on the other hand, might see Elijah in a new light, as a potential father figure. Their relationship could shift from a somewhat adversarial one (due to different loyalties in the show) to a more united front, as they face the challenges of the pregnancy together.
One common theme is rivalry turned friendship. They start off as enemies but over time, in many fanfics, they learn to respect and even rely on each other.
There's a fanfic called 'Elijah's Redemption through Damon'. This one dives deep into Elijah's character development with Damon as an unexpected catalyst. It shows a different side of their relationship, more of a mentor - mentee kind of bond at times.