The character has had some changes since her first incarnation to be sure. Before Barbara Gordon entered the picture there was a blonde that became the first Batgirl. Or rather Bat-Girl. Here's a rundown on her character, the Kanes and wondering where she can go from here.
Rumor has it Bette (Betty back then) and the original Batwoman (her aunt Kathy) we're created to stop the rumors that the dynamic duo were homosexual. Which is rather ironic given the second Batwoman. Kathy and her became the love interests for Batman and Robin. They left regular canon after awhile since they were seen as being too silly.
Bette would show up and like Dick Grayson she used a code name that came from an old Superman story: Flamebird. She was quite the Robin fan and became inspired to be a hero. This had mixed results as she more or less stayed a character that was rarely taken seriously. Her hero worship of Nightwing wasn't of interest to him and much like Stephanie Brown she was told to give up. And much like Stephanie she refused quit.
Her aunt Kathy was murdered although according to Morrison she turned out to be alive and the leader of Spyral. Much of Morrisons' work is questionable in current canon so how much transfers over is unclear. Thanks to him and Batwoman #25 we also have a better understanding of the Kane family tree. Kate and her sister Beth (Alice) are cousins to Bette. Kathy isn't related by blood as she's their aunt by marriage to their older uncle who later died. Their fathers and Kathy's late husband apparently had two other siblings: Phillip who died unknowingly saving his nephew and Martha--Bruce Wayne's mother. Yeah Bruce canonically dated his aunt and he's Jewish on his mothers 'side. Bette and Kate are in fact his cousins which is interesting because their not actually in the inner circle of the bat family. He hasn't had much to do with Bette yet he hide her ID which indicates he at least cares a little.
After getting nearly getting killed Bette was trained by her uncle Jake and became Hawkfire. She hasn't really gotten much respect from the superhero community since although her improved skills were discussed along with her flaws by Jake, his second wife and his friends. I'd love to see more of her and have her fleshed out more. Have her interact more with others, have more of her family life and show her personality. She was once the first Batgirl and should get more respect.