What is noni fruit? Scientifically named Morinda citrifolia, it's a tree in the same family as the coffee plant and is native to Southeast Asia. While it's most commonly referred to as noni, it goes by many other names too including cheese fruit, great morinda fruit, Indian mulberry, and beach mulberry.
In terms of antioxidants, noni fruit is only marginally good for you. With an ORAC rating of only 800, less exotic (and less expensive) fruits test higher including all varieties of apples, blueberries, strawberries, and many more. Therefore if one consumes morinda fruit, hopefully they're doing so for its taste or some other health benefits unrelated to antioxidants.
USDA Database for the Oxygen Radical Absorbance Capacity (ORAC) of Selected Foods, Release 2 - Prepared by Nutrient Data Laboratory, Beltsville Human Nutrition Research Center (BHNRC), Agricultural Research Service (ARS), U.S. Department of Agriculture (USDA) - May 2010