I tired and tired, but couldn't figure out a way to add a hyperlink to the header background image that would link it to home.
If you mean the background-image property in CSS you can use jQuery to select that area and do your stuff, but if you mean the image tag within your header area you can alter it via region--header.tpl.php and add your desired HTML code.
Could you please provide an example?
According to Drupal 7 template suggestions, you can override default templates in your theme. For example, if there is a header region (which is defined in your theme's .info file) template suggestion for this region would be region--header.tpl.php. Look for different variables which are generated inside region.tpl.php and override them as you wish.
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