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She was a war orphan that has found by a female Marine on the battle field. She was just an infant, being carried by another war orphan who was only a few years old. This girl became her adopted sister, Nojiko. The Marine who found them was named Bellemerre. She ran off to her home village and raised these two girls as her daughters.
They didn't have much money, but became a close family. When Nami was eight this gang led by the merman Arlong used their power to intimidate the villages of the whole island. If people wanted to live they had to pay 100,000 per adult and 50,000 per child. Everyone one paid, but they discovered Bellmerre's home. Nami and Nojiko had witnessed what happened in town and ran to try and warn their mother. They were too late. Bellemerre tried to fight back but was easily defeated. Arlong used his strength to crush her arm. Even with her emergency money, she could only raise 100,000. This would be enough for her, but this would mean that Nami and Nojiko would have to try to run away from the island. At this time Arlong's men hadn't seen the girls. The town doctor was trying to hold the girls back in hiding. He told them the plan, but Nami didn't want to leave.
As Arlong and his crew were about ready to walk away, Bellemerre stopped them. She said that money was for her two daughters. She couldn't pay for herself. She said that she loved being their mother too much to ever deny them. Nami and Nojiko overheard this. They escaped the town doctor and ran to their mother's side. Bellemerre told them both to grow up and be strong women. She said she loved them, then Arlong shot her dead before them.
 
That's just the short version of that one story. It gets a lot better when Luffy comes into the story and defeats Arlong for Nami and the whole island. The weird thing is, Luffy never really knew the whole story. The only thing he knew was that Arlong made her cry. That alone was enough reason for him to go and crush Arlong once and for all.