Two-Face appeared perhaps three times in the Golden Age, and was unused for roughly twenty years before his Bronze Age revival. This is, however, a somewhat downplayed example; the character's visual was popular enough that there were several stories with "impostor" Two-Faces published over the '40s and '50s, and the character's Golden Age debut was delivered to future generations in several reprint issues.
The Riddler and The Penguin made more frequent appearances, but were still, at best, recurrent characters. Today, they are regular cast members. The Penguin is now an unshakable Gotham crime boss; Two-Face, the Riddler, and Man-Bat have all Ascended and regularly bounce back and forth between "villain" and "dubious ally" status.
Catwoman is now firmly an Anti-Hero, and rather less "anti" than many of her peers in that group.
Harley Quinn was a one-shot Joker minion from Batman: The Animated Series who was drawn only because they wanted a girl-coming-out-of-the-cake gag. Joker minions have an average lifespan of less than a single episode, but Harley became big enough to not only jump to the comics, but get her own self-titled series as well.'
Probably the biggest one would be Doctor Hurt. Originally, he was an unnamed psychologist involved in a one-off Silver Age story, Robin Dies at Dawn. When Grant Morrison made that story a pivotal part of his run, Doctor Hurt received a name for the first time in 45 years and became the Big Bad of the Myth Arc, the head of the Black Glove, and very possibly Satan, with his experiments in the original story all being part of a long-term plan to destroy Batman.
The tie-in comic for The Batman, The Batman Strikes, used Rupert Thorne and Solomon Grundy far more than the show it's based on did, the former (barring some cameos in Season 4) only appeared at the start of the first episode as the last major crime boss Batman puts away before the Rogues Gallery appear and while the latter's episode ended with the implication he was real after all, "Grundy" in his only episode was really mostly Clayface impersonating Grundy.
Though not a Villain, StephanieBrown, a.k.a the Spoiler, was created to be a plot device by Chuck Dixon for a story that was intended to revamp her father, the Cluemaster, z-list fodder at best, but ended up being so popular that she became Tim Drake's on-again, off-again Love Interest, his temporary successor as Robin IV (though this was as much a marketing scheme as anything else), and Batgirl III with her own Fan-favourite Solo Series. Not bad for a character who began as a plot device, huh?
Bernard played a fairly small role as one of Tim's high school friends in Robin (1993) and mostly disappeared from canon after Tim changed schools, appearing only once. Since being reintroduced in Urban Legends, he's become Tim's love interest and is a major character in Tim Drake: Robin where he's Tim's established boyfriend.
Live-Action TV
Sophie in Batwoman is a main member of the supporting cast, both as the second-in-command of the Crows and as someone Kate still has feelings for. Her comics counterpart had the same role in Kate's Coming-Out Story, and has otherwise made a single non-flashback appearance.
Video Games
Peter Grogan is a very minor character in Batman lore, usually as the police commissioner between Gillian Loeb and Jim Gordon, with the only thing known for sure is that Gordon notes in Grogan's initial mention in Year One that Grogan was even more of a Dirty Cop than Loeb. In Batman: The Telltale Series, he's the commissioner throughout the first season.
In Batman: Arkham Asylum, Clayface appeared as a cameo shuffling between using three other characters' models, unimportant to the story, while Ra's al Ghul showed up as a tagged corpse. In the sequel, Clayface is the Final Boss while Ra's al Ghul is one of three Big Bads.
Harley Quinn was originally just a one-time moll character who made such an impression she became the Joker's pseudo-girlfriend/top henchwoman, then began developing relationships with other characters and got her own spotlight episodes, including a comic tie-in detailing her origins that got adapted into an episode. Then she became a Canon Immigrant into the comic universe and a Breakout Character in general. So she's ascended twice from one-shot to supporting character, from cartoon to comics, in that order.
Robin was featured infrequently in the first season, with the series bible even stating that he was not intended to be Batman's full-time partner like in the comics. Thanks to some Executive Meddling, the show was renamed The Adventures of Batman and Robin in its second season, and Robin appeared in nearly every episode.
Batgirl also appeared relatively infrequently throughout the original series, but was promoted to lead character status when the show was retooled as The New Batman Adventures. Bruce Timm claims her increased prominence was requested by network executives, who felt adding a woman to the main cast would win over female viewers.
A stated goal of Beware the Batman is to utilize obscure Batman villains who haven't been as heavily exploited as his A-list rogues. This means unknown (to the general public) villains like Magpie, Anarky, and Professor Pyg are getting their moment in the limelight, with Ra's al Ghul, Lady Shiva, Killer Croc, and Deathstroke as the only remotely-recognizable villains who appeared in the show (Harvey Dent also appeared, but was unable to fully become Two-Face before it was cancelled). Katana also gets promoted to Batman's principal sidekick.
over the '40s and '50s, and the character's Golden Age debut was delivered to future generations in several reprint issues.