Well, when it comes to naming private stories, you might consider using significant dates. For instance, if the story is related to a particular day, say 'July 15th Encounter'. Also, emotions can be a great source. 'The Heart - breaking Revelation' could be a name. And don't forget about using a key object in the story. If there's a special locket, you can name it 'The Locket's Tale'.
Some private name stories are about nicknames. For instance, a person named Alexander was always called 'Xander' by his close friends and family. This nickname started when he was a little kid. His younger sister couldn't say 'Alexander' properly and it came out as 'Xander'. Since then, it has been a special private name within the family circle that represents closeness and a bit of family humor.
One good private name story could be about a family who named their child after a special place where they had a life - changing moment. For example, a couple who got engaged at a beautiful meadow named their daughter 'Meadow'. It's a private name that holds deep meaning for the family.
One way is through oral tradition. Family members can simply tell the stories about names during family gatherings or when a new baby is born. For example, when introducing the name to the new family member, they can share the story behind it.
You can choose based on the personality you want to give the girl in the story. For example, if she's strong - go for something like Brianna. If she's shy, maybe a name like Violet would work.
You could name it 'My Secret Chapter'. This gives a sense of it being something personal and exclusive, like a chapter in your life that only you know about.