I know that coconut oil is composed of multiple saturated fatty acids, but is there a reason why this would have increased the fitness of the coconut plant?
This is entirely speculation, but the part of seeds that is pressed for oil(the endocarp) is enriched in saturated fats and long fats in ocean-going fruits. The coconut is 91% saturated fats, the oil palm is about 50%, etc. The tamanu or 'ballfruit' produces monounsaturated long-chain fatty acids in the endocarp, about 75% C18 oils, mono- or di-unsaturated.The calabash is astonishingly 20% saturated and 78% C18 oils, and is also sea-distributed(these seeds are called disseminules).
It's not a totally straightforward correlation. Peanut oil is 75% C18+ and 15% saturated and the oil palm mesocarp is only 50% saturated, but the endocarp is 81% saturated fats.
There's an additional consideration here: longer chain fatty acids are slightly more energy efficient per unit mass than shorter chain ones.
Without better data on energy density or spoilage rates on different blends of long-chain fats I can't conclude conclusively that spoilage and energy density are why disseminules tend to have longer-chain oils and more heavily saturated fats, or even that they definitely do. But it looks like they do, and that would make sense.