Feel free to interpret the question as referring to either permanent human habitation or human habitation in general.
Wikipedia has an article called List of countries and islands by first human settlement. The latest by continuous habitation is Crozet Islands, which was discovered in 1772 but was only intermittently inhabited (by sealers) until 1963 with the establishment of the Alfred Faure research station. However it is part of France, which is obviously much older.
This leaves Seychelles, which was first permanently settled in 1770 by France, then Britain, but gained independence in 1976.