Alternative Character Interpretation: Not a full example as this is actually mused upon in the book, but the idea that the serum doesn't actually transform Jekyll into a different-looking evil man, it simply transforms him into a different-looking man, and it's the intoxication of being able to get away with any crime that leads him to act so evilly. Many adaptations, especially recent ones, decide to eschew the idea that it changes his looks at all and represent the changes purely by acting.
In the book Jekyll pretty much admits that his motive for inventing the serum was nothing other than For the Evulz; he wants to be able to act immorally, but as Jekyll he'll always be worried about his respectable image. As Hyde he doesn't have to worry, as the worst people think of him is that Hyde might be blackmailing him (and/or could be his Bastard Bastard son). And keep in mind that Jekyll chooses to keep turning into Hyde, even after Hyde severely injures an innocent child.
A line Jekyll makes about Hyde growing in stature, as though conscious of a more generous tide of blood. Does he mean Hyde would have grown to hulk-like proportions or that he's just becoming healthier compared to the skinnier dwarf form he starts off as compared to the more hearty stocked Jekyll?
And then there's the implied Jekyll/Utterson context as seen below.
Common Knowledge: There are a few widespread misconceptions about the novel. For example, the fact Jekyll and Hyde were one and the same was a twist is not often known (probably because it was so famous), and despite multiple adaptions including this element, there were no love interests in the original story (in fact, there were no named female characters at all).
Harsher in Hindsight: Hyde's crimes were heinous enough, but soon after the book was published, the Jack the Ripper murders took place. Even worse, one of the suspects was an actor who played Jekyll and Hyde onstage; his performance was so convincing that people began to believe it wasn't an act.
It's really easy to read some Utterson/Jekyll into the former's concern for the latter, with his fears that Hyde was Jekyll's son or lover, and was using that to blackmail him.
Homosexual undertones were read into the book early on, and a few of Stevenson's gay friends chided him for possibly bringing them to light at all. The recent passing of homosexual legislation up north meant that closeted homosexuality wasn't just a hot-button issue at the time, but that Stevenson could possibly have had it on the mind while writing. A closer look at the edits from the second manuscript seems to support this theory, as Utterson himself starts to read a little bit more into Jekyll and Hyde's perceived relationship. Then again, this was a time when two men could have a completely platonic Romantic Two-Man Friendship and not be chided for it.
A short story by Kim Newman, "Further Developments in the Strange Case of Dr. Jekyll and Mr. Hyde", riffs on this; Essentially, Hyde is a separate person. And he's Jekyll's lover.
It Was His Sled: The original story is a mystery about what connection the upstanding Jekyll could have to the shady Hyde. Pretty much everyone nowadays already knows the answer; Jekyll and Hyde are one and the same, the result of Jekyll taking a potion that split him into two selves, one normal and one totally evil. Often enough, the twist is the only thing they know about the story.
Most people don't know that the dual identity was originally a Twist Ending, and it is not uncommon to see adaptations of references where Hyde is a hulking monster rather than simply an evil, apelike man.
Misaimed Fandom: Some people will use the idea of being a "Jekyll and Hyde" as an excuse for either their own bad behaviour or that of their loved ones. This arguably inverts the moral of Stevenson's story, where Jekyll's refusal to take responsibility for Hyde's actions was a big part of what caused things to go badly.
Moral Event Horizon: Clearly, when Hyde brutally murders Sir Danvers Carew for absolutely no reason, he has reached this point. Even Jekyll denounces him. However, Jekyll's decision to keep changing into Hyde despite the horrific consequences and harm it has done to others that he is fully aware of is one in and of itself.