I remember seeing her in one of the later episodes. She was the nurse to the guy who stole Fonzie's cool in a nightmare. It was one of those terrible post-Richie episodes.
When there's no more room in Hollywood, remakes shall walk the Earth.

I remember her as the drummer too. :)She was great in Satan's Cheerleaders from 1977 too. The whole movie is on YouTube.

Welcome to IMDB. You are obviously knowledgeable about Happy Days. There are many threads on this board that you could comment on. I prefer conversing with fans on IMDB over exchanging pleasantries with strangers and casual acquaintances on facebook. There is no need to use your real name on IMDB. There are a couple wackos who harass and annoy people on these boards. I am not joking.
An extra appearing multiple times on the same show is not uncommon. The only one I can think of at the moment is Richard Couture on "What's Happening!!". That guy was at school, at the malt shop and anywhere Raj, Dwayne and Rerun happened to be.
These are Hillary Horan's Happy Days appearances according to IMDB:

• Welcome to My Nightmare (1981) Nurse Elaine
• Fonzie, Rock Entrepreneur: Part 2 (1977) Daphne
• Fonzie's Baptism (1977) Girl Dancing with Ralph (uncredited)
• Spunky Come Home (1977) Peggy
• Fonz-How, Inc. (1977) Girl in Red Scarf (uncredited)
• Joanie's Weird Boyfriend (1977) Girl Playing Drums (uncredited)
• The Graduation: Part 1 (1977) Brunette Prom Girl in Blue (uncredited)
• Marion Rebels (1977) Girl in Red Jacket (uncredited)
• A Shot in the Dark (1977) Girl Playing Drum (uncredited)
• The Book of Records (1977) Girl Playing Drum (uncredited)
• Fonzie's Old Lady (1977) Girl Playing Drums (uncredited)
• Richie Branches Out (1976) Fonzie's Brunette - Green Skirt (uncredited)
• A.K.A. the Fonz (1976) Cheerleader on Fonzie's Left (uncredited)
• The Muckrakers (1976) Brunette in Yellow Top (uncredited)
• They Shoot Fonzies, Don't They? (1976) Girl Playing Drums (uncredited)
• Fonzie's Hero (1976) Girl in Red Jacket (uncredited)
• Fonzie the Father (1976) Brunette in Red Jacket (uncredited)
• A Mind of His Own (1976) Birdhouse Girl in Green (uncredited)
• Fonzie Loves Pinky: Part 3 (1976) Brunette in Orange Skirt (uncredited)
• Fonzie Loves Pinky: Part 2 (1976) Girl Playing Drum (uncredited)
• Fonzie Loves Pinky: Part 1 (1976) Brunette in Yellow Jacket (uncredited)
• Arnold's Wedding (1976) Brunette in White (uncredited)
• A Sight for Sore Eyes (1976) Brunette in Blue and White (uncredited)
• Two Angry Men (1976) Girl in Maroon Scarf (uncredited)
• Fonzie the Superstar (1976) Audience Member in Purple (uncredited)
• Tell It to the Marines (1975) Dancer in Red (uncredited)
• Fonzie's New Friend (1975) Girl in Red Scarf (uncredited)
• Three on a Porch (1975) Girl in Red Scarf (uncredited)
• A Date with Fonzie (1975) Girl in Plaid Skirt (uncredited)
• Jailhouse Rock (1975) Brunette in Grey Skirt (uncredited)
• Richie Fights Back (1975) Girl in Red Skirt (uncredited)
• Fearless Fonzarelli: Part 2 (1975) Majorette (uncredited)
• Fearless Fonzarelli: Part 1 (1975) Girl Playing Drum (uncredited)
• The Motorcycle (1975) Short-Haired Brunette (uncredited)
• Fonzie Moves In (1975) Brunette in Plaid Skirt (uncredited)

There was never a show called 'Hilary Horan '.

Her imdb page stops at 1982. I'm guessing she probably retired from show business, started a family, etc. I'll always remember her as the girl doing the drum roll and crying just before Fonz jumps the garbage cans. There are 5-6 background actors you see pretty much in every episode, at least until the sixth season or so.