It's easy to selectively save Gmail image attachments to Google Drive, but if you want to automate the process, tech blog Digital Inspiration shares a script that makes the archiving happen automatically.
The script runs in the background of Gmail. When you receive a new attachment it automatically sends it to a folder called "Gmail Images" in your Google Drive. It also scans your email to pull out images from older emails so you don't have to go through those thousands of older messages.
If you want manual control, this script does the same thing but gives you full control over what gets sent over. You can grab the auto-save script, as well as installation instructions over on Digital Inspiration.
Auto-Save your Gmail Image Attachments to Google Drive [Digital Inspiration]