Beach Blanket Bingo! Released in 1965, it's part of the series of the beach party films starring Annette Funicello and Frankie Avalon. Ah, teen screwball comedies, even in the 60s you were one of a kind. Anyway, while the main focus of the movie are the romantic hijinks between Annette and Frankie's characters (Dee Dee and Frankie, respectively), there is a wee little subplot around one of their friends, Bonehead.
Now, see, Bonehead isn't the sharpest tool in the box, but he's got a kind heart and a sweet demeanor. As this is a beach party film the ocean gets quite a bit of screentime, and at one point Bonehead almost drowns during a surfing attempt and, wouldn't you know it, he is rescued by a mermaid. The mermaid's name is Lorelei, and she explains that she's been alive for centuries but never known a decent guy like Bonehead before. A sweet, if sadly brief, summer romance follows.
It's a cute little subplot in a fun movie, predating Madison's appearance by quite a few years. We get to see Lorelei's blue tail in a scene where she and Bonehead introduce themselves properly (sorry, I couldn't get a good screencap of that bit) and later Lorelei shows up with legs, which she has temporarily gained in order to try human clothes and dance human dancing (and apparently get her hair done in a human salon). But like many summer romances, this ends on a bittersweet note, though both characters have plenty more good things to look forward to in their futures.
Thanks to Gemma and Shira for independently telling me about this film!