Is there a way I can store my application-specific data on Twitter's end? If I have a configuration value to save and the user logs in via Twitter's OAuth API, can I post some data and save it specific to that user?
Similar to: https://developers.google.com/drive/web/appdata
No, currently Twitter doesn't support the storage of any application-specific data unless this data are tweets, lists of users or any other resource that could be written by an API call.
If you want to add data to a specific user you are going to need to use other storage and use User id as the foreign key to find the data related to a specific user.