Ladybirds are named after Our Lady, the Virgin Mary. The red colour is said to represent the red cloak Mary was depicted wearing in old paintings. The seven spots are for the seven joys and seven sorrows.
They were called 'ladybugs', too, in olden times and that is, I believe, still the most common name for them in America. Maybe people here thought that associating Our Lady with the word 'bug' was somehow not quite right!