She and Lucy's father were originally part of a merchants' guild called 'Love & Lucky' where they two met and decided to become independent when Layla became pregnant with Lucy. They chose her name when they noticed that the sign for the guild had the letter K missing so that it read 'Love & Lucy'; it made an impression on them so they decided to name their daughter after it.
In her home, Lucy kept a good relation with the staff in the estate and presumably her mother until she passed away when Lucy was around the age of ten on 7-7-X777, the same day that the dragons disappeared. She was 29 when she died.