I like Dropbox more than Google Drive. Currently when there is an attachment in GMail, there are two buttons - download and add to GDrive. Can I somehow replace or add a Dropbox button?
Dropbox has released an extension for Chrome that allows you to save attachments to Dropbox and to insert links to Dropbox files into new messages. Here's the link to the extension:
And here's a link to the Dropbox help article detailing the extensions and how to troubleshoot it if something goes wrong:
No, "download and add to GDrive" copies the file between Google servers without downloading it to your computer. That's why it's built in to Gmail.
You can download the file to your Dropbox folder on your computer. (You could save steps by putting a Dropbox-folder shortcut where your "Save" dialog can use it. This varies by your desktop OS.) But you still have to save into your Dropbox folder.
For Gmail to implement "download and add to Dropbox" directly (without going through your local computer), Gmail would need the ability to log in to your Dropbox account.
(You might try it on Android. On Android, both Dropbox and GDrive register as storage services, and you can remain logged in to both of them. So when Gmail or any other app downloads a file, it could let the user pick where to store it.)
Not sure whether this is still relevant, if not just ignore.
Once you open the file from the gmail or the inbox app in Android, click on the three dot menu at the top right of the screen (just below the clock on the notification bar). From the pop-up menu select "send file". This will show you the list of apps on your device to which the file can be shared. Select Dropbox from that list and you can go through the normal steps in Dropbox to select where the file is to be saved.