SEOUL—Fans of pop stars send their idols all sorts of crazy stuff. In South Korea, the true measure of a fan's devotion is sending lunch to the stars.

On a recent Saturday here in Seoul, members of A Pink, a seven-member girl group, worked their way through beautifully crafted lunch boxes featuring egg salad with tomatoes, basil and toasted breadcrumbs, rice wrapped in seasoned lettuce, and beef-and-vegetable-skewers, all paid for by a fan.

"We...