From 1989 to 2010, Barbara's role was to be a character that found new ways to be a hero despite suffering an injury that ended her career as a "normal" crimefighter. She served as a way to show that heroes could come in different forms, and when partnered with Black Canary she was part of a unique crimefighting duo - the "brains and brawn" dynamic, but with some twists you didn't see elsewhere.
She also played a role in keeping the various Bat-books connected, and her injury helped make it clear that Gotham was a "grounded" part of DC's line of comics. This was (mostly) back when Denny O'Neil was the head editor of the Bat-books, trying very hard to keep the Bat-books a little island of relative normalcy, where there were no easy solutions. That's not to say that the Bat-books had no sci-fi/supernatural elements during his tenure, but they tended to be things that were ridiculously dangerous and came at a high price even if used right. Super-science was more likely to turn you into an abomination than cure your ills. When there was a major earthquake or a viral outbreak, the Justice League were not there to just wave their hands and make the problems disappear. The wider DCU was there, but it was a thing for the occasional fun "Flintstones meet the Jetsons" crossover, not something meant to change the normal status quo in Gotham.
After 2011 her appeal is, well, >>126557774
said it best. She's yet another hot young extremely fit hero, who happens to be a genius. Oh, and her dad is friends with Batman.
I suppose you could argue that her chief role these days is to remind us all that the big companies own a lot of characters that would be better off in the public domain.