THANKS TO PASITEA, AN IRRESISTIBLE NYMPH who lured in Poseidon, the dreamy seaside village of Positano was discovered. It has six beaches to choose from. There’s the large, busy Spiaggia Grande, where you can watch handsome fisherman glide in and out. A short walk away is the quieter Fornillo beach, where you can enjoy drinks and delicious meals on the porch of Hotel Pupetto.

But for the feel of discovering your own hideaway and a delicious lunch, head to Arienzo. It’s a small cove bordered with giant rocks, with views of fishing boats bobbing along the horizon, ferries headed for Capri and, in the distance, the Li Galli islands. Legend says these islands were once mermaids whom Ulysses turned to stone so they’d stop trying to seduce him off his course.

I’m not talking great sand. It’s volcanic and coarse with lots of black pebbles. But like almost everything you touch around here, Positano’s black pebbles have a story behind them. If you find one with a hole in it, it means the BVM passed through it, and it’s blessed. You’ll see many Positanesi wearing necklaces of these black pebbles.

The sand situation means you should bring along beach shoes. Reserve in advance for your lounge chair, umbrella, and little table where a spritz can be delivered. Then get totally comfy, lie back and get lulled by the lapping of the calm water.

Over a decade ago, Melody, an American who had lived in Positano for years, tipped me off that Ada’s gnocchi at Arienzo was famous in these parts. My first lunch there, on a quiet day in early October, remains tops in my memory books.

These days, Arienzo is no longer a local secret, but it’s still family run, and despite its popularity hasn’t lost its homey feel or deliciousness. The beach restaurant is set up on stilts, looking like something Thurston Howell III would have built on Gilligan’s Island, perched to take in the view. Along with Ada’s gnocchi, the menu features spaghetti with clams, Caprese salad, grilled fish caught that morning and granitas—flavored ices made from Positano lemons and whatever else is in season. During summer season, reservations are a must to get in on this paradise.

Even if you don’t find a pebble with a hole blown through it, the moment you arrive at Arienzo Beach you’ll feel Positano’s blessings.

Arienzo Beach (www.arienzobeachclub.com): The best way to arrive is via a small boat from Spiaggia Grande, or you can walk down a zigzag path of 300 steps from the Arienzo bus stop.

Golden Day: Visit Arienzo Beach for sun, sea, and lunch. To keep the celebratory vibe, have dinner at Mediterraneo ristorante (www.mediterraneopositano.com) to enjoy fantastic seafood, fun service, and a Neapolitan guitarist who strums classics.