There are six days to go before Thanksgiving, so Josh Pettit’s Mooresville, North Carolina, home isn’t yet adorned with a single Christmas decoration. But it sure looks like Santa’s been here recently. On the dining table, the 47-year-old web developer has neatly arranged a vast assortment of gifts — everything from T-shirts and handmade blankets to stuffed animal toys and baby dolls — that have recently arrived at a PO box for his mother, Betty Pettit, an Alzheimer’s sufferer. All of them were sent by complete strangers. “Lately there’s been an onslaught of gifts,” Josh says, marveling at Bet...