Personal Chronicles. This name gives the idea that it's about your own personal stories that are only for you or a select few. It has a bit of a classic and timeless feel to it.
Hidden Gems. It implies that the content in the private snap story is something special, like precious gems that are hidden away from the public eye. It could be used for snaps that include unique memories, private jokes, or exclusive events that only a select few are supposed to see.
How about 'Behind the Lens'? It implies that there are hidden or unseen aspects of your life that you're sharing. Or 'Snapshots of Me' which focuses on it being about you and your experiences.