I'm assuming her lore will be a contrived, plot-hole filled, cliche and poorly written short story summarizing the reason she joined the league, her personality, her vague motivations and her open ended goals. That way, there will be no need to develop her character in the future and she can just remain static forever. (No, I'm not sore about Soraka, why do you ask?) I took a stab at writing her lore following those guidelines:
Nami was the royal body double for her kingdom's princess in an aquatic kingdom underneath the waves. She was loyal to the crown and did all in her power to protect it. One day, as Nami had been asked to stand in for the princess as it was too "boring" during a royal meeting, disaster struck. The meeting room was raided by a group of people conducting a coup. Nami had been trained in the ways of aquatic combat since she was very little and did her best to fend off the invading marauders, but she was held down by multiple fins. The king, a just and kindly old man refused to fatally wound any of the intruders as they were people he knew personally in the kingdom and, in front of Nami's eyes, he was slain with his own trident. Nami cried out in vain as one of the marauders wearing a mask slaughtered every delegate, official and representative at the meeting with the king's trident. The masked intruder then menacingly stepped towards her holding the King's trident, readying a killing blow. Suddenly, a warm light emanated from the deceased king's heart and flew into Nami's. As she felt the surge of power in her fingertips she realized that she was, in fact the true princess as only those with the blood of royalty could possess such a power.
She used her power to pin every one of the intruders to the walls with intense waves of water as she interrogated them for information. Reclaiming the King's trident from the masked marauder and removing the mask, Nami realized that it was in fact, the princess who was underneath the mask. The princess admitted to doing so because she had found out that she would not be able to become Queen as Nami had been the true successor all along and was simply learning the life of the common people to have a better rule over the people. Not pleased with this, the princess organized a rebellion, getting the help of her adoring public to be on her side.
Realizing that she could not kill the princess because she loved her like a sister, nor any of the other intruders as her compassion was too much, she escaped the kingdom she had called home with heavy sobs. The Princess announced that Nami had killed the King and spears were being thrown at her by the people, believing her to have killed the king with the blood soaked trident in her hands.
Eventually after months of swimming and barely surviving she washed onto a shore where she was greeted by Fizz, an old acquaintance who seemed extremely happy to see her. Fizz introduced her to the league and she joined so that she could hone her skills, gain allies and one day shed light on that fateful day to her people and take it back from the clutches of the power mad princess,
"As the one and true princess, I demand my loyal subjects venture from this kingdom. We shall make new homes elsewhere and conquer the sea, and eventually, the land! Together we shall avenge my father and the people! Nami and all those who stand by her will be brought to justice!" - The Imposter Princess