St Lucia was a young Christian girl who was martyred, killed for her faith, in 304AD. The most common story told about St Lucia is that she would secretly bring food to the persecuted Christians in Rome, who lived in hiding in the catacombs under the city. She would wear candles on her head so she had both her hands free to carry things. Lucy means 'light' so this is a very appropriate name.
December 13th was also the Winter Solstice, the shortest day of the year, in the old Julian calendar and a pagan festival of lights in Sweden was turned into St. Lucia's Day.