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.
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.