When it comes to knitting, Lydia Schoenbein thought she had seen it all. The 73-year-old retired nursing-home supervisor from Morton, Ill., learned to knit and crochet when she was growing up in Germany and can make everything from socks and shawls to cable-knit sweaters.

So when her 22-year-old granddaughter, Carly Hill, visited and showed off her own creation—a crude-looking, ropelike cream scarf—"I was flabbergasted, to tell you the truth," recalls Ms. Schoenbein. The accessory...