THAT is why feeding Pudge was so important to Lilo! She lost her parents to bad weather. A storm caused her happy family to break and fall apart. So with her childish reasoning she created the idea that if she could appease this little fish that controlled the weather she could prevent another bad storm that would cause someone else to lose any loved ones.
The name "Lilo" means "Generous One" and its origin is Hawaiian. It can also be interpreted as "Lost" and this would give the song title "He Mele No Lilo" a loose translation as "Lullaby of the Lost". The name Nani means "Beautiful" in Hawaiian.