A short name, like
customerDB, is perfect. Because you'll use it in code, this ID must contain only alphanumeric characters with no spaces.
As the name suggests, Private IDs are never shared with other users. If you grant another user a permission to query a dataset you own, then they will use their own separate Private ID to reference that dataset.
If you are the dataset's owner, then you must set the Private ID when you create the dataset using Dataset Manager. You can also use Dataset Manager to change it later, though of course you'll then need to change any code you've written which referenced the old ID.
Alternatively, if you have been given a permission to query a dataset owned by someone else, then a Private ID is automatically generated when you accept the permission. If you want to, you can use Dataset Manager to change it to something more descriptive.
See also public name and description.