The Royal Hawaiian was purchased by the Sheraton Corporation in 1959.
When Gidget and her parents check-in at their hotel, the sign behind the desk clerk says Sheraton Hotel. Later, when Mr. Lawrence receives a telegram reply from Jeff, it's addressed to "Russell Lawrence, Royal Hawaiian Hotel"
Throughout the film, several exterior doors (from the Lawrence home in California to the hotel rooms in Hawaii) open outward as opposed to inward as exterior doors typically will.
Gidget and her parents argue about the timing of the trip to Hawaii soon after they announce it at the breakfast table. But neither parent mentioned when the trip was planned for.
In an early scene, Jeff gives Gidget his fraternity pin. We know this is in "real time" because when her parents spring the Hawaiian trip on her Gidget complains she's only just been pinned. But Jeff pinned her at the end of Gidget (1959). Further confusing the matter, we soon see the whole getting pinned scene reprised shortly thereafter as one of the flashbacks Gidget confides to Abby on the airplane.