Named after Luigi Giglio Lilius - Italian doctor, philosopher, chronologist (unkn-1576).
Lilius is a lunar impact crater that is located in the rugged southern highlands. Just to the southeast is the slightly larger Jacobi. The outer rim has been worn down and rounded, particularly to the northwest where the inner wall protrudes farther into the interior floor. (wikipedia entry)
A crater is a circular depression likely created by an impact event. On the Moon they are named after deceased scientists, polar explorers, astronauts or cosmonauts.