Attachments.me integrates your Gmail account with your other cloud accounts on Google Drive, Dropbox or SkyDrive.
Once you install the Attachments.me extension, it adds a little upload button in your incoming Gmail messages that have attachments. You can click the button to upload the file attachment – be it an image, video or a document – directly to any of your linked cloud accounts.
You can later go to the attachments.me website or use their mobile apps to search through all your Gmail attachments visually.
Attachment.me lets you create rules such that matching attachments from incoming messages are automatically saved in pre-defined folders. Also, in addition to saving attachments, the services also lets you insert files from other cloud services into your Gmail messages.
Update: Attachments.me is no longer available. A good and free alternative is Google Scripts.