The easiest way to replace the "Gmail Add to Drive" button with a "Dropbox Add to" button is to use a browser extension. I recommend the "Button for Dropbox" extension for Chrome.
Once you have the extension installed, open Gmail and click on the "G" icon in the top-right corner. Then click on "Settings" and scroll down to the "Send mail as" section.
Click on the "Add another email address" link and enter your Dropbox email address. Then click on the "Save Changes" button at the bottom of the page.
Now, when you compose a new email, you'll see a "Dropbox Add to" button instead of a "Gmail Add to Drive" button.