Islands of the North Atlantic
One feature of this name is that the acronym IONA has the same spelling as the island of Iona which is off the coast of Scotland but with which Irish people have strong cultural associations. It is therefore a name with which people of both main islands might identify. On the other hand, it can be confusing (a) because of this duplication of the name of the existing Iona and (b) because Greenland, Iceland, and Newfoundland are also important and rather prominent islands of the North Atlantic, which the term is not intended to include.
"British Isles" remains for now the most widely used term to describe the aforementioned territories, though people to whom it seems anglo-centric may have difficulties with the term; while accurate in describing both the geography and the politics of the islands when Ireland remained part of the United Kingdom of Great Britain and Ireland (1801-1922), it has not changed to reflect political developments since 1922.
It remains to be seen whether IONA, which has been used as part of the Northern Ireland Peace Process, will become a widely accepted replacement term for the British Isles, whether another term will evolve over time, or whether the status quo will prevail.
References
[1] Denis Canavan MSP, British Irish Inter-Parliamentary Body, Summary of the 15th Plenary Session, 9. The Future of the Body
[2] Trevor Sargent TD, "The Good Friday Agreement", Dáil Éireann speech
[3] Mr. Peter Luff MP, British House of Commons speech
[4] Kevin Barry, "Resiliency, Tolerance and Avoidance in Northern Ireland"
[5] Taoiseach Mr Bertie Ahern TD, "Ireland and Britain A New Relationship for a New Millennium"
[6] Paul Sharp, "When New Meets Old: Irish Diplomacy, Northern Ireland and the Peace Process"