The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at You Know, That Thing Where.
Death By Child Birth, Missing Mom, You Gotta Have Blue Hair (assuming that she is a human and her hair is naturally blue), Purple Eyes (assuming they indicate her powers), Bi The Way, Superpower Lottery, Superpower Meltdown and/or Power Incontinence- it depends on what you mean by 'unstable'.
Without knowing how she's related to the villain and how they view each other, best I can give is Dysfunctional Family Index; what you're looking for is probably in there.
I'm trying to find some tropes my OCs fit in for my original series, and the descriptions I am giving you are extremely brief. For example, the main character is the daughter of the general of the royal army, and her mother died during childbirth. She has blue hair and violet eyes, and controls one of the most important powers in the story. She is bisexual. She is related to the major villain of the story, and her powers are very unstable.