There are different kinds of batting, and some are more likely to "beard" than others. Wool, especially the natural batts you can have made at the mill, are prone to bearding (also called fiber migration). Cotton isn't usually a problem, but the needlepunched cotton is sturdier than traditional cotton, which wads up into little cotton balls inside the quilt.
Polyester batts come in a thousand brands and styles. A needlepunched poly batt won't beard as much as a "traditional" poly batt (that's what it says on the package - "Traditional".) Glazed or bonded battings are also a good choice. If you are hand quilting, the glaze or bonding makes it slightly more difficult to needle. Not significantly. The label will tell you how closely you have to quilt the batt. That is a good indicator. If you have to quilt it very closely, it is more likely to beard than one that can be quilted less densely.
As Purplemem said, you can get black batting.
Using good quality cotton fabric is important. I don't mean to sound like "Quilt Police", but there is a reason for this - the fibers in polyester are sort of hooked - they catch onto each other. A poly-cotton backing and a poly batt, rubbing together, will sort of meld together. I am not saying you used a poly blend fabric - it's just something to keep in mind when shopping.
Usually, bearding first occurs where the needle punches holes. Change needles very frequently. A sharp, fine needle makes a smaller hole and isn't as likely to push and pull the batting through the hole. This is true for machine and hand quilting. On a queen size quilt, even if I don't break them, I go through at least a package of needles (and I do break another package of them).
Hope that helps in the future... I know it doesn't do you much good now.