Personal Chronicles. This name gives the idea that your private snap story is like a personal diary, chronicling your unique experiences, thoughts, and feelings that you only want to share with a select few. It has a bit of a nostalgic and personal touch to it.
Well, 'Behind the Scenes' is a nice option. It gives the idea that there are things not usually seen. Another one could be 'Personal Paradise', suggesting a special and private place for you.