Dyeing for Your Art: Boris Karloff removed his partial bridgework to achieve the monster's creepy sunken-cheeked appearance. In Bride of Frankenstein the monster learned to speak, preventing Karloff from repeating the effect; as a result, the monster looks considerably less weird the second time around.
The reason that Victor Frankenstein's first name was changed to Henry (which was his best friends first name) was because the producers felt that Victor would sound too unfriendly and severe for Great Depression audiences (but maybe because they made Frankenstein's best friend Henry Clerval a bit of a severe character, they decided to change his name to Victor.)
The studio got cold feet over several aspects of the film. Reaction to a preview screening resulted in a mandate to add an introduction warning audiences about what they were about to see; the death of Maria was considered too graphic so it was cut (resulting in audiences thinking something even worse had happened to her, robbing the Monster of much of the viewer's sympathy); a key line about Henry saying he knew what it felt like to be God was censored for decades; and an ending was tacked on to reverse the original plan to kill off Henry at the end.
I Am Not Spock: Boris Karloff subverted this by admitting that the Frankenstein movies are the only reason he became a successful actor.
Bela Lugosi was originally set to play the Monster. He also created his own monster make-up, which is reported to have resembled the eponymous being from the 1920 film The Golem. 20 minutes of test footage was shot with him, but it is considered to be lost. Lugosi reportedly hated playing the character so left the project alongside the film's originally assigned director. Lugosi got his chance to play the Monster 12 years later in Frankenstein Meets the Wolf Man. It could be argued that his instincts were correct as he is widely considered to have been miscast as the Monster. Before this, he was also offered the role of Dr. Frankenstein...which would've made for an especially amusing Casting Gag, as the character's mentor is played by none other than Dr. Van Helsing, while his assistant is Renfield!
The role of the Monster was also offered to John Carradine, but he refused it because he felt that it was beneath him. Carradine later played Dracula in a couple of the lesser later Frankenstein-franchise films.