You might consider 'Exclusive Snap Tales'. The word 'exclusive' makes it seem special and only for a select few (in this case, just you). Also, 'tales' gives it a bit of a story - like feel which is great for a collection of snaps. For example, if you're using it to document your travels, each snap is a part of the travel tale. So it works well for various types of content that you might want to keep private on Snapchat.
Well, to create a private story on Snapchat, first launch the app. Then, tap on the 'Create Story' option. After that, there should be a privacy setting where you can select 'Private' to keep it only for the people you choose.
Some nice names could be Lily, which represents purity and beauty. Another option is Sophia, meaning wisdom. Also, Ava, which has a modern and sweet feel.
Some other options might be 'Hidden Chapters', 'Private Musings', or 'Confidential Chronicles'. These names give the sense of something that is only for you or a select few and is not meant to be shared widely. They convey a sense of privacy and exclusivity which is what a private story is all about.
You could name it 'My Secret Moments'. It gives a sense of the story containing personal and private experiences that are just for you or a select few.