My Intimate Recollections. This name is great because it emphasizes the intimacy of the stories. When you think of recollections, it's about memories, and the 'intimate' part makes it clear that these are personal and private memories that are being shared in the private story.
For an Instagram private story, you could use 'Giggle Gallery'. It makes it seem like a collection of things that will make people giggle. Another one is 'Smile Snippets' which gives the impression of short, happy moments.