Private Moments Gallery. This name gives the idea that your snap story is a collection of personal and private moments that are precious to you. It has a bit of an artistic and exclusive feel to it.
You can also consider names like 'Exclusive Insights', which gives the idea that it's something only a select few can see. Another option is 'Private Perspectives', as it implies that it's your own view of things and is private. 'Confidential Chronicles' is also a great name as it has a sense of secrecy and a story - like quality.