And a year later, apparently DropBox integration for attachments is now built into iOS 9 - but it doesn't work from inside DropBox like you would think.
FIrst you have to have DropBox installed on your IOS device.
Next you must enable DropBox as a file location for attachments. Open a new Mail message and long-press in the blank message body. From the popup menu you will need to choose Add Attachment (on iPad) or More and then Add Attachment (on iPhone)
The default popup selection window should be your iCloud files, but there will be a link on the upper left named "Locations." Click that link.
Click the "More" link in the list.
Click the buttons to turn on Dropbox and any other cloud services you have installed that you want to enable as attachment locations, such as Google Drive or Microsoft OneDrive.
Start a new Mail message and long-press in the body, then click Add Attachments as before.
Now when you click the Locations link, Dropbox should appear as a clickable item in the popup list. Clicking on that should open a file picker window into your DropBox account where you can select files to be attached to your Mail message.