Oh, look, it's a pretty Hollywood heroine who is in love and faints. How original.
(To be fair to Elizabeth, she faints because her fiancé is a psycho who has reanimated a corpse…and then the reanimated corpse menaces her. We'd faint, too.)
But even with her very extenuating fainting circumstances, there's not much to Elizabeth as a character. She spends the whole film talking about how much she wants to get married, taking a break only to faint when the monster comes after her. If she ever thinks of anything other than Henry, you don't get to hear about it.
She's kind of…dull.
And Henry seems to know she's dull. He keeps putting off his marriage and the happy Hollywood ending, in order to go run around collecting dead bodies and hanging out with Fritz. Elizabeth says:
"The day we announced our engagement, he told me of his experiments."
It's like Frankenstein is using the experiments in order to put off dealing with his engagement. And of course, his most important, passionate relationship in the film isn't with Elizabeth; it's with the monster.
It's worth pointing out here that director James Whale was gay. He never married; instead, like Frankenstein, his most important emotional relationships were with other men, especially with the Hollywood producer David Lewis.
You could see Frankenstein, then, as being all about how Frankenstein doesn't want to get married. Poor Elizabeth, in this reading, isn't really a person at all; she's just a stand-in for life that all men circa 1931 were supposed to lead. She symbolizes the life that Frankenstein's running away from.
Elizabeth doesn't get to be an interesting character because Frankenstein is about men hiding away with men—and how that leads to tragedy and panic in the village.