When Cindy asks Richie who his favorite singer is, he says Margaret Truman. He's referring to the daughter of President Harry S Truman. She was an operatic soprano in the 1940s and '50s who performed on radio and TV, at recitals, and with orchestras.
Both Ron Howard and Linda Purl, who plays Gloria in this episode, played a child of Andy Griffith in two different hit TV shows for him. Howard was Sheriff Andy Taylor's son Opie on The Andy Griffith Show (1960), while Purl was attorney Ben Matlock's daughter Charlene during the first season of Matlock (1986).
Linda Purl plays Gloria in this episode. She became a cast member during Season 10, where she would play Ashley Pfister, Fonzie's steady girlfriend.
The record Richie (Ron Howard) breaks in Cindy's (Cheryl Ladd) hotel room is by Elvis Presley. A (45 rpm) record would have one song on each side. This record had the Elvis songs "Love Me Tender" on one side and "Jailhouse Rock" on the other side. Both songs were also movies with Elvis Presley.