Sicily, Sardinia, Corsica, Crete, Cyprus.
Are versions of these the first names we have for them, or do some have older unrelated names?
The precise etymology of Corsica is uncertain, but it was also known to the ancient Greeks as Σειρηνούσσαι (Seirēnoússai).
Crete is referenced in Mycenaean Greek ca. the 15th century BCE as /ke-re-te/ and /ke-re-si-jo/ (I can't find the Linear B versions at the moment). However, there are references from 300 years earlier to a place called /kaptara/ in texts from Mari (in modern Syria). The nearby city of Ugarit mentions a place called /kabturi/ around the same time. Contemporary with the Mycenaean references, the Egyptian pharaoh Amenhotep III's Aegean List (an itinerary from a diplomatic expedition) refers to Crete (or at least to a location on Crete) as 𓂓𓏺𓆑𓍘𓏲𓈉 /kꜣ-ftỉʷ/ (*Kaftōˀ). Much later, the Bible mentions "Caphtor". These may all refer to an older Minoan endonym for the island, or at least for part of it.
Cyprus also has another 15th c. BCE Mycenaean reference as 𐀓𐀠𐀪𐀍 /ku-pi-ri-jo/. The Amarna Letters from the same period refer to the island (or to the dominant kingdom on it) as *Alašiya.
The ancient Greeks called Sardinia Σαρδώ (Sardô), and tied it into a legend about a woman from the city of Sardis in what is now western Turkey. However, the name is both older and probably local to the island, maybe even an endonym from the Nuraghic civilization that inhabited Sardinia during the Bronze Age. As early as the 13th c. BCE the Ancient Egyptians refer to a people called 𓆷𓄿𓏭𓂋𓏺𓂧𓄿𓋔𓄿𓈉 /šꜣʲʲ-r-dꜣnꜣ/ (*Šardana) as mercenaries, pirates, and later among the "Sea Peoples" associated with migrations during the Bronze Age collapse. Egyptian depictions of the *Šardana match up nicely with Nuraghic depictions of their own boats and warriors. Later, when the Phoenicians arrive on Sardinia, they make a reference to people called /šrdn/ on the Nora Stone.
The ancient Greeks called Sicily Τρινακρία (Trinakría) after its three headlands. The current name comes from Latin Sicilia, which was derived from the name of the Sicels (Latin: Siculi ; Greek: Σικελοί / Sikeloi), a local tribe. The Sicels may also have been referenced much earlier in 13th-12th c. BCE Egyptian texts as 𓆷𓄿𓎡𓏺𓃭𓆷𓄿𓌙𓀂𓏪𓈉 /šꜣk-rw-šꜣ/ (*Šekulša), who are mentioned as also being part of the "Sea Peoples", though this may instead refer to people from Sagalassos in southwest Turkey.