In most cases HTML body of the message references attached images using special "cid:" protocol. It specifies the Content-ID of the image, not the name of the file:
This is our <strong>brand new</strong> logo: <br />
<img src="cid:logo@example.com" />
The actual image is embedded inside an email, as a part of a MIME tree, as an element related to the HTML body and with content-disposition header set to inline.
You should use IMail.SaveHtmlAs method. It saves HTML version of the message body as regular HTML file (changing cid: references). All visual elements are saved as files in the same folder.
If the message is plain text only, this method uses GetBodyAsHtml() to create HTML from plain text.