Selecting File from the menu bar brings up the following menu.
The orange draw.io logo in the top left will take you to www.dropbox.com. Clicking the title of the diagram allows you to rename the Dropbox file.
This page will look at the menu options circled in the above diagram, as these are not common to all storage options. Other option categories, such as creating and opening files, importing and exporting diagrams, or using libraries, are common to all storage options, and are explained elsewhere.
This page will also explain how autosaving is handled when using the Dropbox storage option.
Saving and copying files
Selecting File -> Save, saves the currently open file. It is saved to whichever storage option is currently selected - in this case Dropbox.
In order to change the name of the saved file, or to switch to using another storage option, use File -> Save as and change the fields in the resulting menu screen before clicking Save.
File -> Make a copy duplicates the file in a new tab, and allows you to change both the name and the storage option for that file, just as with File -> Save as.
However, there is also the option to Decide later. Should you choose this, the menu screen will close, allowing you to work on the copy. However, because no storage option has been selected, you will be asked to make a selection via the Save as menu screen when you later save the document.
Dropbox files are autosaved any time a change is made to the diagram (you can switch this off under File -> Document properties). If you have a native client with notifications of file changes enabled, this can be annoying. In this case switch off the autosave and perform manual saves when required. Dropbox does not, at the time of writing, provide a mechanism to avoid the notification from triggering during autosave.