In Buddhist belief, if someone has reached enlightenment, what will become of them when they die?

It depends. If they have chosen to become a bodhisattva, they will be reborn as human to help guide others to enlightenment. This is sometimes viewed as the more compionate choice, as you continue to help others. If they have not chosen to become a bodhisattva, they simply cease to be, forever rid of desire and self, one with everything.