Despite appearing only briefly at the end of season 1, Lexi Branson, and her subsequent death, were actually pretty important to the plot of the CW's Lexi Branson's only live appearance on the vampire drama occurred in the eighth episode of season 1, entitled "162 Candles." Lexi is visiting Stefan for his birthday, but ends the evening shockingly dead. Aside from the fact that Lexi has earned an impressive following despite being short-lived, Lexi's death actually made a significant impact on season 1.
To be more specific, and a vampire that appears in "162 Candles.". When she first meets Elena, she's alarmed because of her resemblance to Katherine, but when the confusion is cleared up, she's ultimately very supportive of Elena's relationship with Stefan.
Lexi's murder was far from a senseless act in season 1. In fact, the shocking twist was actually part of a greater trend in the series which was to establish that no one in Mystic Falls is safe.
Another similar situation is when . Vicki is Matt's sister and Jeremy's love interest in season 1. She has a pretty rough go of it during her time on show as she is attacked and mentally manipulated by Damon, and ultimately, turned into a vampire. In season 1, episode 7, the episode before Lexi's death, Vicki is killed by Stefan after she attacks Elena and Jeremy.
Along with raising the tension and fear throughout season 1, Lexi's death was vital to the plot. Throughout season 1, Stefan and Damon try to avoid any attention being drawn to them, especially when it comes from law enforcement. However, around the time of Vicki's death, suspicions are high and Damon, as per usual, enacts a morally ambiguous plan to save himself and Stefan: frame his behavior on Lexi so that the police no longer suspect him.
Additionally, Lexi's death continues the rising tensions between Stefan and Damon. At this point in the show, the brothers are definitely in a sort of rivalry wherein Stefan is the good vampire and the hero and Damon is the morally grey and dangerous monster. Lexi's death only emphasizes these points. Plus, Damon murdering Lexi is a betrayal of Stefan as well because of their close friendship. Therefore, Lexi's death keeps these themes moving until they finally explode at the end of the season.
Although Lexi had a very small role in she has managed to become a fan favorite over the years. One big reason for this is that Lexi returned several times after her death in later seasons of thereby deepening her character and her backstory. Namely, Lexi's return showed that, not only was she Stefan's friend, but she also was his closest confidante when it came to turning his humanity on and off. Lexi consistently helped Stefan keep his humanity, and even helped Damon despite his difficulties.
Overall, it seems that has gained so much popularity because she is a kind and generous person. While most vampires can be ruthless and cruel, Lexi shows that it isn't too difficult to be a good vampire, and she continues to be this even after death. Her connection to Stefan is an admirable one, and even though she was a minor part of as a whole, her goodness made her shine through.