Update any embedded attachments/images that use http. This one updates the src attributes that use double quotes:
UPDATE wp_posts
SET post_content = REPLACE(post_content, 'src="http://www.yoursite.com', 'src="https://www.yoursite.com')
WHERE post_content LIKE '%src="http://www.yoursite.com%';
This one takes care of any src attributes that use single quotes:
UPDATE wp_posts
SET post_content = REPLACE(post_content, 'src='http://www.yoursite.com', 'src='https://www.yoursite.com')
WHERE post_content LIKE '%src='http://www.yoursite.com%';