I am not an expert on saving local files. but I do a lot with databases.
As far as i know, the whole idea of writing to local files on mobile is considered a major security threat, so pretty much the only local option would be to use Local Storage. You could take an array full of data and store that in Local Storage easily. I guess it depends on how much data you want to store, and how important data integrity is - if it is local, people will find a way to modify it in ways you might not want.
Setting up a database on a web site and passing data back and forth with AJAX calls is not that hard and works really well for me. Then you have access and a lot more control over all the data your users are saving - if you want to do any kind of statistics, or checking if anything suspicious is going on.