It's already assumed that Joss Whedon's Batgirl movie will focus on the Barbara Gordon version of the character. That makes sense, as she's the longest-serving incarnation, in addition to being the most popular version (and also, the current comic book Batgirl) - but Babs is far from the only Batgirl in DC comic book lore. If Whedon wanted to look elsewhere, what other Batgirls are available to choose from?
Introduced in 1961's Batman No. 139, Betty was the niece of Kathy Kane, who was spending her evenings at the time fighting crime and attempting to woo Batman while disguised as Batwoman. After discovering her aunt's dual identity, she managed to get involved in the family business as Bat-Girl, but it wasn't to last; after just seven appearances, Betty was quietly dumped from the Batman series altogether. She'd later reappear as a member of the Teen Titans in the 1970s, and again in the 1980s, where she adopted a new superhero name: Flamebird.
Bertinelli's tenure as Batgirl was an intentionally short one: Already an established DC character when she adopted the identity, after debuting in 1989 as the Huntress in The Huntress No. 1, Bertinelli became Batgirl in 1999's Batman: No Man's Land No. 1 during a rare period where Batman was absent from Gotham City, working on the assumption that criminals would be more scared of someone wearing the bat than they would any other hero. It worked for a brief time, before her already strained relationship with Batman came to a head over her inability to follow orders and an overly violent approach to crime fighting - she was fired by the Dark Knight upon his return to the city. She returned to the Huntress identity afterwards.
The daughter of two assassins, Cain debuted in 1999's Batman No. 567 as a child trained from an early age to follow in her parents' footsteps. However, she turned her back on them after her first murder, and instead, became the ward of Barbara Gordon, who eventually convinced Batman to help train her. As a result, in the same year's Legends of the Dark Knight No. 120, she became the latest Batgirl. She eventually abandoned the role after a confrontation with her mother, although she returned later in the new role of Black Bat. Currently, in the rebooted DC comic continuity, she goes by the name Orphan.
Another child of a bad guy - this time, the somewhat lame villain Cluemaster - Brown (who first appeared in 1992's Detective Comics No. 647) was initially known as the costumed vigilante Spoiler, before being recruited by Batman in 2004's Robin No. 126 to become the first female Robin in DC's main timeline. (Carrie Kelley had predated her in 1986's Batman: The Dark Knight Returns, but that was officially outside of the regular mythology of the character.) After faking her own death, Brown disappeared for a number of years before returning as Batgirl in 2009's Batgirl No. 1. Following the New 52 resetting of DC's comic book history, she was restored to her Spoiler identity.
Technically, Tiffany Fox hasn't become Batgirl yet. She made her first appearance (of only two, to date) in 2014's Batgirl: Futures End No. 1 in a flash-forward appearance, set five years from "today." In that alternate reality, Fox is one of three Batgirls in a team known as the League of Batgirls - the others being Cassandra Cain and Stephanie Brown - controlled by a mysterious figure calling herself the Bête Noire … who turns out to be none other than Barbara Gordon. The League operates outside the law, but with the same aim that Batgirl always had: keeping Gotham City safe from whoever, or whatever, seeks to cause trouble.