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.
Well, 'Personal Pics Parade' makes it seem like a fun collection of your personal pictures in the snap stories. 'Secret Snap Selections' gives the idea that you've carefully chosen these snaps for a private viewing. And 'Private Photo Pile' is like a little stash of photos that are only for a special group or just for you. These names can make your private snap stories more interesting and unique.
Some cute names might be 'My Hidden Highlights', 'Exclusive Exposures', or 'Intimate Insights'. These names give a sense of privacy and something special that is only for a select few to see. They also sound friendly and inviting in a way that makes your private snap stories seem like a little treasure trove of personal moments that you are choosing to share in a more secluded way.