In the history of ancient Greece, Ionia was the region of western Anatolia, in the Aegean. What's the relationship to the Ionian Sea, the body of water to the West of mainland Greece? Someone in /r/mapporn referred to islands in the Ionian Sea as "Ionian Islands," which I guess isn't technically wrong, but I personally think of Samos, Chios etc.
The names are unrelated.
Iōniē "Ionia" refers to the land of the Iōnes "Ionians" (and in Bronze Age Greek the long ō was originally āō, so Iāōn-).
The Ionios sea, with a short o, is of unknown etymology, though some ancient writers believed it was named after Io swimming across it. Derivations from ios "violet" or ios "arrow" might also be plausible, but there's no evidence to suggest them.
"Ionian islands" is certainly ambiguous in English (but not in Greek). I'm afraid we'll just have to live with that.