The short answer: late May, June, September and early October. The long answer depends on which island group you are going to, and what you want out of the trip. This page pulls together the month-by-month picture across all eight archipelagos.
Sea temperature, by month.
The Mediterranean warms slowly and cools slowly. May sea is still cold (17 to 19°C) for anyone not from northern Europe. June is 20 to 22°C. July and August are 24 to 26°C, peak swimming. September holds 23 to 25°C. October drops to 21 to 23°C, still swimmable for most northern-hemisphere adults. November is 19°C and dropping. December to March: 14 to 17°C, wetsuit territory.
Best months, by island group.
Sicily: October is the honest answer. September a close second. See the full Sicily month-by-month.
Sardinia: September wins outright. Sea warm, crowds gone, prices dropped. See Sardinia month-by-month.
Bay of Naples (Capri, Ischia, Procida): May and early June before the summer crush; late September and early October after.
Aeolian: May, June, September. July and August have the yacht scene and the crowded Stromboli boat tours. Winter closures on Filicudi and Alicudi.
Egadi: May to October. Sea holds warm into November on the southern coves.
Pontine (Ponza, Ventotene): June, September. August is Roman-holiday chaos.
Tuscan (Elba, Giglio, Capraia): May, June, September. Peak Italian domestic tourism in August.
Pelagie + Pantelleria: May to October. The southernmost islands stay swimmable longest. Pantelleria’s Passito wine harvest is late August into September.