Lampedusa has a genuinely North African climate: hot, dry summers and mild winters. The sea is warm enough to swim in from May through October, and at its best in June and September when the water has had time to heat up but the August crowds have not yet arrived.
July and August are peak season — the island fills with Italian holiday-makers, prices double, and Rabbit Beach can feel uncomfortably crowded. If you can go in late June or early September, you'll get the weather without the pressure.
Spring (April–May) is lovely for walking and birdwatching — Lampedusa sits on a migration route between Africa and Europe, and hundreds of species pass through. Winter is quiet to the point of sleepy: many restaurants and hotels close, but the island has a certain bleak beauty and the locals are happy to chat.