Preheat oven to 350°. Spread coconut out on a small rimmed baking sheet and toast, stirring occasionally, until golden, about 5 minutes; let cool.
Combine rice, ginger, salt, and 1¾ cups water in a medium saucepan. Bring to a boil, reduce heat to low, and cook, covered, until liquid is absorbed and rice is tender, 30–35 minutes. Remove from heat and discard ginger and lemongrass; fluff rice with a fork. Serve rice topped with toasted coconut.
Combine cabbage, mango, black rice and vinaigrette in a medium bowl and toss to combine. Adjust seasoning to taste.