Enforced Method Acting: Sean Connery is truly terrified of spiders. The legs of a stuntman were used when the tarantula crawled onto Bond. Note the shot when the spider is on his shoulder — it's walking on a pane of glass. Once you've seen it you can't unsee it.
No Budget: United Artists only lent $1 million. The art director got some extra cash from the producers' own pockets, and the special effects people convinced UA to give an extra $100,000 to do the Collapsing Lair. In 2014 dollars, this amounts to about $8.5 million. For comparison, the budget for Skyfall was $200 million.
Peter Burton as Q (although he's only identified by Q's real name, Major Boothroyd). He was unavailable for the second movie due to taking a role in another film that was never completed (a decision he regretted for the rest of his life), and Desmond Llewellyn would take over the role starting with From Russia with Love.
To a lesser extent, Jack Lord as Felix Leiter. Lord wanted a bigger role, more money, and co-star billing to appear in the next movie, which was declined, and Lord was too much in demand for leading roles to return to what was effectively a bit part. Leiter would be played by a succession of other actors in subsequent films.
Same Language Dub: Strangways and the photographer are dubbed, and Nikki Van der Zyl dubbed over Eunice Gayson and Ursula Andress...
Tuckerization: This film marks the first appearance of the character Major Boothroyd (later played by Desmond Llewellyn and known simply as "Q"), based on fan and gun expert Geoffrey Boothroyd who informed Fleming the gun he had James Bond using was a wimpy ladies' pistol. (This also formed the basis for Bond's forced gun upgrade in this movie.) The real-life Geoffrey Boothroyd appears in a vintage featurette on the Dr. No DVD and Blu-ray, in which he demonstrates the relative stopping powers of the two Bond guns, plus his own favorite, a Ruger .44 Magnum.
Bond's original line was simply "I am James Bond", but Sean Connery felt it was rather weak and went off script, resulting in one of the most famous movie lines of all time. For the first few takes, he said "Bond, James Bond" without pausing. It was only when he paused for just a split second to light a cigarette between the "Bond" and the "James Bond" that the magic happened.
Director Terence Young, who once had turned Lois Maxwell down on the grounds that she looked like she "smelled of soap, not sex", offered her either Moneypenny or Sylvia Trench, but she was uncomfortable with a revealing scene in the screenplay. She ended up playing Moneypenny.
The first draft script strayed quite far from the book. In this version, Dr. No was a monkey god worshipped by native islanders, and the human villain was someone else. Saltzman and Broccoli were unsatisfied with this approach and got Richard Maibaum to rewrite it to be closer to the novel. The original screenwriter was not pleased with the rewrite and had his name removed from the script. Maibaum would have a hand in writing all but three Bond movies through License To Kill.
A number of different actors were considered for the role of James Bond. Cary Grant was a strong contender, but he would not commit to more than one or two movies, and Saltzman and Broccoli already knew they were going to make at least five or six. (Grant would later take advantage of the Bond-inspired craze for spy movies by playing a debonair spy in Charade.)
Potential actors to play Dr. No included Fleming's next-door neighbor Noel Coward, who famously responded to a telegram asking him to take the role with "Dr. No: No, no, no!" (and who later visited the set with Fleming, nearly disrupting shooting), and Fleming's distant cousin Christopher Lee (who would finally get his chance to play a Bond villain in The Man with the Golden Gun).