Elizabeth, in Mary Shelley's Frankenstein, represents the typical woman who "resides" in the private sphere (the domestic sphere). Victor shelters her from his studies, even leaving his (their) home in order to separate her from science.
Essentially, Elizabeth's job is to care for the Frankenstein family. After Caroline's (Victor's mother) death, it is up to Elizabeth to insure that the younger children are cared for. In fact, when she falls apart after William's and Justine's deaths, Victor's father begs Victor to come home so he can restore happiness to her.
The only women the reader comes into contact with are Elizabeth, Justine and Caroline. This said, the only settings we see them in are ones in which they are surrounded by, and devoted to, the family (even Caroline's death shows Elizabeth at her side).
Therefore, the women in the novel are the backbones of the family (meant to nurture and care for the family); they are not meant to take part in the world of the man (the world of math and science).
Elizabeth is a very passive character in the text, alike to any other woman that is mentioned within 'Frankenstein'. This causes the reader to comprehend the roles of women vs. the roles of men; women are seen as less capable, they don't endeavour the unknown, they stay within the known, and keep at the obvious.
However, Elizabeth is connected to Frankenstein through a great devotion and love that she feels towards him. It is not just because she was told she would marry him some day, she actually does have a connection with him. They seem to understand each other well, and she and him really care about one another. We notice this when the daemon kills Frankenstein's Elizabeth, and he can finally not hold on anymore, he has finally had enough. He now, more than anything seeks revenge of the monster and does not care what will happen to him in the process.
Elizabeth and Frankenstein are also connected in the way that they grew up in the same house, almost as if they were brother and sister, and they therefore grew around the same people and experienced the same events.
Hopefully this helps you... :)