'Secret Scribbles'. It gives the idea of a private collection of thoughts or events written down. Another could be 'Hidden Chronicles' which implies a series of events that are not meant to be widely known. And 'Personal Parables' also works well as it suggests personal stories with a bit of a moral or lesson, but still private.
A simple yet effective name could be 'My Private Musings'. It's straightforward and gets the point across. It tells people that this is a place where you jot down your personal thoughts and experiences that are not for public consumption. It's like your own little mental diary in the form of a private story.
How about 'My Mysterious Memoirs' or 'Confidential Chronicles'? 'My Mysterious Memoirs' has an air of secrecy and personal history, making it a great name for a private story. 'Confidential Chronicles' also conveys the sense that it's a record of something private and should be kept confidential.
You could name it 'My Secret Chronicles'. It gives a sense of mystery and makes it clear that it's something personal and only for a select few to see.
'The Unspoken Chapters'. It gives the sense that there are parts of your story that haven't been told or shared publicly. 'Solitary Sagas' is also a great name as it implies that these are stories that are solitary, meaning private and only for you. And 'My Quiet Quotients' which can be seen as the personal elements or stories that are quiet and private.
Friendship's Hidden Hideout. This name shows that the private story is a hidden place where the secrets and special moments of your friendship are stored, away from the rest of the world.