We chose the Arenella as a quiet hotel with its own little beach, despite its bad Tripadvisor press - mostly criticising its director and service. The truth was in all ways the opposite. We did not get the room we thought we had reserved; the staff and later the director were charming and apologetic, but the hotel was full so we could not move rooms, so we agreed to stay one night and review the situation next day. We took our bags up the windi8ng steps to our room and went to explore the beach. It is a 15 minute walk down a steep road crowded with cars. The beach is indeed small (and sandy) but it, too, is crowded with people. You can hire beach chairs and umbrellas for about 20€ per day; the hotel doesn't even provide beach towels..We walked back, dejected. We had been told that the hotel had no restaurant and had no problem with that because we had a car and it is only about 3Km to Giglio Porto. But Arenella now has a vegan restaurant which really was quite tasty.
That night, a family and their baby decided to have an all-night party on the terrace outside our bedroom. Before breakfast (good buffet with ham and cheese and the usual breads and pastries, but terrible coffee) we decided to leave.
We decamped to the Hermitage (I'll review it in a few minutes).
My advice is to avoid Giglio altogether if you want peace and a beach, unless you stay at the Hermitage. And if you do that, don't waste 100€ bringing a car over to the island. Arrange to leave it in Pto San Stefano.

Just out of Giglio Porto about 2 miles is a side road to a small rocky beach. On this, the Arenella Road, is the Hotel Arenella, perched on multiple levels high above the sea. The hotel and the rooms have a marvelous sense of simple modern Italian design. The views are magnificent. The short walk to the beach is down a steep paved road, or you can drive or take a bus (1 mile away is the stop) or a taxi to a lovely sandy beach at Giglio Campese. The breakfast is simple but adequate. The staff are welcoming, in Italian and English. The only restaurants are in Giglio Porto and Giglio Castello, with some beachside simpler places in Giglio Campese. So you do need to eat in town. The hotel has a navetta (shuttle van), and you can bring a car over on the ferries out of season. Traffic becomes restricted and impossible in July and August.