A large portion of Liz's personality is simply to keep other people at arm's length. When she lets her defenses down, she is warm, kind and caring. Not many people get to see this side of her, but it does show through every now and then.
Another facet to Lizbeth Locke is her sultry side. She can be incredibly seductive; typically saving this side of her personality for women she finds attractive. Unlike some, she does have a good sense of what's appropriate and what isn't in a given situation, and is quite practiced in the art of the 'slow and gently' seduction. It has often been said that she could have any person in the universe that she desired... and it's completely true. The only question (in her mind) is how worth it would it be?
She also has an interesting tie to both life and death. She can sense the presence of living being nearby, often giving her advanced warning. While it is possible for her to differentiate individual lifeforms, it requires concentration, and is not something she often has time or inclination to do.
The alternate history is actually the one she lived, or at the very least remembers living. She was Simon's older sister. They were quite close, until their younger sister, Alicia, died. Lizbeth took the death very hard, and left Terra. She headed out towards Freelance space, and eventually ended up getting herself killed at the age of 25.
Simon took the loss of his older sister hard enough when she left, but when she died... he refused to accept it. So, he had her body cloned, most of her memories transferred over, and then left her body in cryogenic storage, until he could explain his actions to their parents. The cost of this procedure was the reason he was living in the relative squalor he was when he met Sylvia.
Lizbeth remained in stasis for 200 years. Simon did collect her body, and ensured she was safe even after he was abducted by the Overseers. The reason it took him 200 years was simply that he wasn't allowed to awaken her until that time by the Overseers. They then took her in, trained her, and she is the person we know today.
It is difficult to tell just where these alternate histories converge. Lizbeth herself sometimes becomes confused, and as such would rather just ignore everything that's not happening now.