Big-Lipped Alligator Moment: The Light Show at the Pyramids of Giza, with the mystical narration. Though in fairness, this is a case of Aluminum Christmas Trees. If you're at the Pyramids of Giza at night, that light show will be playing. That said, the whole Sound and Light Show (which is still playing today, with the same narration) is basically an hour-long Big Lipped Alligator Moment.
Ensemble Darkhorse: Jaws, who along with Oddjob, Red Grant and Baron Samedi is one of the most popular villainous henchmen of all time. The guy became so popular that he's so far one of the only two James Bond villains that were allowed to return to a sequel and get a redemption, and when people hear the word 'Bond Villain', he's likely one of the examples that will be brought up first.
Anya Amasova gets a lot of love from the fandom as well, given her beauty, brains and hyper-competence to the point that many consider her to be Moore's best Bond girl, if not one of the best overall. Even the producers liked her so much that they asked Barbara Bach to reprise her role on two separate occasions, although sadly she declined.
Ho Yay / Worthy Opponent: M and General Gogol. After British and Soviet forces decide to work together, Gogol calls M "Miles", M calls Gogol "Alexis", and they act overly friendly to each other. Bond and Anya give a WTF look to each other after witnessing this scene.
Memetic Bystander: The guy drinking wine when the Lotus surfaces at the beach. Not only does he reappear in the following two films, he's slowly developing a meme.
Rooting for the Empire: Richard Kiel has recounted how Jaws' survival at the end was met with giant applause at several screenings. He might be a villain, but he's such a Determinator that after a while you sort of root for him.