I've always thought that a Mary Sue is a character who has no flaws. Which would make her "perfect" I guess, but IMO it's not about what's perfect about her, it's more about the lack of what's not perfect about her.
Does Serah show any 'bad' personality traits at all? I haven't watched the streams so I only have the trailers and XIII for reference, and she does seem a little... flat. Did she ever lash out on her sister? Be jealous? Racist? Angry? Whiney? Depressed? I have this feeling that no. That doesn't mean I don't like her, it just makes her a little... uninteresting.
The reason I loved Lightning so much was that they intentionally gave her flaws. They decided to take out parts of her personality and hand them to Fang, to make her more serious and bitchy. Does Lightning come across as overly emo? Some think so. I just thought the flaws was what made her believable, a true human character. It gives her purpose and motivation, which is so vital to a story line.
As humans we do have a lot of 'negative' emotions, and to me, for a story to be good, the characters has to do 'bad stuff' for me to believe in them, because we all have a dark side. We all have some conscious or sub-conscious motivation for the stuff we do.
For example, Light being so hard on Serah in XIII was totally believable to me but others might see that as 'unnecessary bitching'. I guess that's part of SE's challenge too, they have to write stories that fit a 14 year old as well as a 40 year old, which can't be easy.