Trait is another word for personality trait, and personality traits define who or what you are like as a person. For example, some people can have very argumentative or agreeable personalities, making argumentativeness and agreeableness their traits. People can also be brave, ambitious, boastful, or even arrogant. The list goes on and on. So in looking for Olivia's personality traits, what we are looking for is what she is like as a person. We learn a couple of her traits in the very first act.
One thing we learn about her is that she is a very emotional person, which we see through her excessive mourning over her brother. She claims to be in such deep grief over her brother that she plans to be in mourning for the next seven years, never accepting company and never showing her face in public except veiled, as we see Duke Orsino's servant explain in the opening scene:
The element itself [Olivia], till seven years' heat,
Shall not behold her face at ample view;
But, like a cloistress, she will veiled walk. (I.i.28-30)
However, since she completely drops her idea of prolonged mourning once she falls in love with Cesario, one can wonder if her decision to mourn is truly genuine, or just a way of avoiding society, especially Orsino.
A second character or personality trait shown in the first act is that she is a very caring person. We can see her caring nature in the fact that, despite wanting to reject all society due to mourning, she permits Sir Toby and his friend Sir Andrew to remain in her house, even though she strongly objects to Sir Toby's drunkenness and wild behavior. We especially see Olivia's objections to his behavior expressed through her maid, Maria, who declares to Sir Toby, "By my troth, Sir Toby, you must come in earlier o' nights: your cousin, my lady takes great exceptions to your ill hours" (I.iii.3-5). However, regardless of how much Olivia disapproves of Sir Toby's behavior, she permits him to remain with her, showing us just what a caring and nurturing nature she has.