From the experts:
Demi appeared in the US as a baby name thanks to a single celebrity, actress Demi Moore, who put it on the Top 1000 throughout the 1990s. As her star faded, so did the visibility of the name, though now it is rising again on its own steam. It's popular in Europe, too, especially in The Netherlands. Demi may be that unusual name that's launched by a celebrity and then maintains its visibility all by itself.