The way I see Olivia is mostly as a device rather than a character, to be honest. She does not have the same depth as Viola, for example (although it is kind of fair to compare them). She is not there to move the story forward. Her presence (passively) allows the story to move forward, but she exists largely in relation to our other main characters.
I think Olivia can also be read as a foil to Viola. Their names are almost perfect anagrams of each other, and for a writer who loved wordplay as much as Shakespeare, this cannot be coincidence. In some ways, Olivia starts off representing in some ways the essence of what womanhood was “supposed” to be, but at the same time she undermines these ideas as we see her behaving mischievously and boldly.
I do prefer Viola to Olivia, but they can both provide an interesting study!
