wasn't just any murder...her friends and her are on a hunting trip and come across 2 guys burying a body...they then proceed to brutally murder her friends, she escapes and is left all mentally messed up and becomes completely agoraphobic as well as just generally afraid that the killers will eventually find her.
Thats why The Ring Of Volthoom (aka the Power Ring from Earth-3) chooses her....because she is utterly fearful. blah blah blah fast forward she FINALLY overcomes her "Great Fear", that ring goes bye bye and a real GL ring chooses her...making her Green Lantern of sector 2814.6. Eventually her and Simon are forced to share a single power battery so that they will become partners and learn to work together.
Shes like Kyle, in that shes a very "Human" GL. John, Guy, Hal...theyre all theses big, strong, smart, rage-y, HERO types...Jess and Simon are written much more Flawed and Down To Earth...thats one of the reasons i really like Her, and to a slightly lesser extent Him (but he does have one of the best designed GL costumes ever, Coming in just below the Justice League 3000 costume for me.)
TLl;DR: Yeah, shes got a pretty messed up origin story in the comics as well as the animated film. Even more gruesome in the books. =]