The Twilight series has been beloved since its debut. Edward and Bella are the starring couple, overcoming the woes of having a vampire-human relationship and eventually finding happiness as a married couple with a hybrid daughter.
It's difficult to picture Edward being with someone other than Bella, but if things had been different, he had other choices of significant others. In another lifetime, or even before, or after, he met Bella, Edward could have found a romance with someone else.
Maggie
Maggie was a member of the Irish coven with the distinct gift for being able to tell the difference between a truth and a lie. She was among the vampires present supporting the Cullens against the Volturi.
Her drive for justice and honesty would no doubt be appealing to Edward, and she too has a family in her coven, alongside Siobhan and Liam. Edward also clearly loves brunettes, so it's a win-win.
Benjamin
Rami Malek plays Benjamin, a member of the Egyptian coven, with special abilities over the elements of earth, air, fire, and water. Benjamin is clearly a good guy and stands up for what he believes in, even going against his creator Amun's wishes to do so.
Benjamin also had a good relationship with Renesmee, using his abilities for her amusement. While neither he nor Edward is canonically shown to be interested in men, and both have significant others, in another life they could have been a great couple.
Rosalie
In another life, if neither Rosalie nor Edward had been turned, it could have worked. What if the two had met as humans? Rosalie wanted nothing more than to be a wife and mother, and given audiences saw Edward as a happy husband and father with Bella, there's no doubt he would've been that for Rosalie.
The two had trouble getting along when audiences were introduced to them, and it has been that way since. However, they do share a mutual dislike for their vampirism and long for something more.
Zafrina
Zafrina is a member of the Amazon coven, and she possesses the power to make others see illusions. She ends up being a supporter of the Cullens against the Volturi.
Edward and Zafrina would make a good couple. Opposites would attract in this case considering their different lifestyles, but Zafrina is great friends with his daughter Renesmee and is thoughtful and considerate beneath her tough exterior.
Angela
Angela was shy, sweet, and intelligent. If Edward had been unable to hear her thoughts, as was the case with Bella, perhaps Edward would've already been dating Angela by the time Bella arrived. Angela's caring disposition and her respect for others are noteworthy qualities that Edward no doubt would find attractive and vice versa.
Angela had goals and dreams though, so when it inevitably came time to choose to remain human or become a vampire, Angela may not have been able to make the commitment. Then again, never say never.
Rachel Black
Tanaya Beatty played Jacob Black's sister Rachel, who made her appearance in the first half of Breaking Dawn. One of the members of Sam's pack, Paul, ends up imprinting on her, and she returns his affections, much to Jacob's annoyance. The thing is, Rachel comes from a familial background of the supernatural and was dating a werewolf.
She may not be up to dating a vampire right off, but she's at least open to the idea of dating someone of the supernatural variety, and wouldn't it be funny to see Jacob irked at the idea of his sister dating Edward?
Emily
Under different circumstances, Emily might have been the "vampire girl" instead of the "wolf girl." Even after Sam had hurt her, she stayed with him, trusting him to not lose his temper and hurt her again, despite the everyday reminder on her face. She embraced the supernatural, not unlike Bella, and she was beautiful and kindhearted.
If she and Edward had met, perhaps they would've hit it off and audiences would be reading a different version of the Twilight series.
Huilen
Of all the vampires that arrive in support of the Cullens, Huilen is the only one that truly understands their situation. She watched her sister die giving birth to human-vampire hybrid Nahuel, but she took him in as her own and raised him anyhow.
If something ever did happen to Bella, Edward might pursue a relationship with Huilen, considering they can relate on so many aspects, especially with Huilen having Nahuel and Edward having Renesmee to think of.
Alice
The idea isn't out of the question. Alice and Edward get along wonderfully. They're both classy and sophisticated, and they both bear supernatural abilities on top of their vampire qualities. They can relate on a variety of levels. They would have made a dashing couple, had Alice not been led to Jasper, and had Bella not moved to Forks years later.
The two were apparently destined to be more like brother and sister than a couple, and audiences will never know what could have been.
Tanya
Tanya would've been a perfect mate for Edward. Like the Cullens, she and her coven lived a "vegetarian" lifestyle, and they considered the Cullens family. Tanya even previously expressed interest in dating Edward, but he turned her down.
Still, the two maintain a friendly relationship, and she does support them against the Volturi. Despite that though, it still seems like Edward missed a grand opportunity to find love, especially with someone he shares so much in common with.