There is not much to say about Jack's sister, some people say her name is Emma, but I haven't find anywhere that could confirm this theory, so for me she's the unnamed sister.
From as much we can see from the short clip of Jack's memories, is that while he loved to play tricks on her, Jackson loved his sister dearly, so much that he gave his life to save her. We see him taking care of her, playing with her and making her laugh. It's the relationship between an older brother and sister - from what we see it gave me the sensation that Jack was protective of her, but still made fun of her, because this is the way we older siblings do, this is how I feel with my younger brother. It's the sensation when Jack can make fun of her, but god forbid it someone else does it.
Whenever I see the part with the ice that is breaking under her skates, and
how Jack tries to calm her down and make her come to the safe part, I tear up,
because it's the where we are shown that whatever we have seen before with how
Jack is, he is a Guardian, he cares deeply and for a loved one he is capable
of great sacrifices.