I know of no "big hits" she has had, stumbled upon or not.
As far as I have been able to figure, her star began rising when she got a gig on a local (San Jose, CA) TV show called (IIRC) People Are Talking, and her "folksy, down-to-earth" shtick caught on with the locals.
She then started making one-off appearances on shows about ghosts and other paranormal subjects, and parlayed all of the above into getting an appearance on The Montel Williams Show, on which she became a weekly guest on what Williams called "Sylvia Browne Wednesdays". Williams was thought of as a stand-up guy, and viewers evidently figured that if HE thought than Browne was "the real deal" (something he repeated almost every Wednesday), then she MUST be truly psychic. Her weekly appearances were little more than show-length commercials for her books and public appearances, with her doing "readings" for the studio audience as well as for some relatively well-publicized guests (such as Shawn Hornbeck's parents and Amanda Berry's mother).
And that's how a cheap carny act (apologies to carny acts everywhere) became a New York Times Best-selling author and a multimillionaire.