I would use poly.
Also, wanted to point out that there are other reasons to use cotton than what the guy from Superior mentions. For example, using 100% cotton fabric, thread & batting (no scrim) makes for a heavier quilt than quilting with poly thread -- especially as I now prefer to quilt at 3/4" spacing. Cotton also fades, whereas poly does not. I've seen older quilts that were quilted with both cotton & poly in the same quilt. The cotton thread fades to roughly the same color as the fabric. The poly thread remains the original color. For heirloom-quality quilts, I still believe cotton is the way to go. The final issue rests with the care of the quilt. If the recipient is likely to be good at following directions, and it isn't an heirloom quilt, cotton or poly both will work equally well. In fact, modern, high-quality poly threads probably would be an advantage. However, for those (like some college kids or even people from a certain generation that believe "it must be washed in hot water to get it really clean") who tend to abuse their quilts by washing in hot & drying on high, I would stick to cotton. The reason is that poly becomes damaged at lower temps than cotton. Short of leaving cotton in the dryer for 20-30 minutes or longer after it's completely dry, you're not likely to damage the cotton fibers. The fabric might crinkle, but that's the worst of it. The only time I've seen poly "cut" (and no, that is not the right word) cotton fabric has been after the owners have dried it on high for years. It may be a combination of the cotton becoming damaged from the heat of the dryer plus the way poly responds to heat by melting & then reorganizing back into a solid shape. Properly cared for quilts do just fine with either poly or cotton threads. Many quilt artists have been using poly threads in their wall quilts for decades without issue.
So, long story short. I'd go with poly thread. It is stronger & will marry together all those thick layers nicer than a cotton would. But really, you can't go wrong with either one for everyday quilts. Do keep in mind, though, that poly threads tend to be shinier than cotton. Depending on the look of your quilt, you may or may not want that effect.