Ponza is an island destination mostly frequented by Italians (especially Romans and Neapolitans, as it's just a hop, skip, and a jump away by boat) that is largely undiscovered by foreign tourists. Ponza is all about the beach-and-boat life, so there's not much "culture" here, but you will find one strange, amateur, interesting museum that you can visit - for free.
When you spend the day at Cala Frontone beach (accessible via a short boat ride from the port of Ponza), take a break from the scorching sun, and head up the steep stairs of the cliff to the restaurant and museum (both with a fabulous view!). The museum (with no name, just called "museo") is filled with all sorts of collected objects from the history of Ponza. The handwritten signs are only in Italian, but you'll appreciate the various statues of Ponza's patron saint San Silverio, work tools, photos and paintings of the island, old keys, books, furniture, musical instruments, letters, and more. A sleepy dog guards the museum, resting out of the sun on the cool whitewashed floors.