Here are some of the details: Support for Web Storage API
My first thought was some sort caching mechanism. Like, if I have a script that fetches data from the internets or takes a very long time to produce some piece of data, I could use localStorage to save that data. Then if someone executes the script again in a very short amount of time, I could give them the cache in localStorage.
My second thought was some CRUD stuff. Something like zoxide came to mind. I could make (and I actually did) a glorified CRUD utility that stores the path to the most frecuently visited directories, and then query that and use the result to "jump" quickly to that location.
Beyond that I'm having a hard time imagining how this could be useful. I'm not saying it's useless, having a tiny database can be handy. I just don't know when to use it.