Some nice names could be Lily, which represents purity and beauty. Another option is Sophia, meaning wisdom. Also, Ava, which has a modern and sweet feel.
Some other options might be 'Hidden Chapters', 'Private Musings', or 'Confidential Chronicles'. These names give the sense of something that is only for you or a select few and is not meant to be shared widely. They convey a sense of privacy and exclusivity which is what a private story is all about.
You could name it 'My Secret Moments'. It gives a sense of the story containing personal and private experiences that are just for you or a select few.
Personal Tales Unveiled. This name gives a sense of sharing personal experiences that are only for a select few. It implies that there are stories that are unique and not meant for the general public, which is perfect for a private story.