The image that's used for a URL for the Like, Send, or Comments function is defined by the Open Graph tags. JFBConnect can try to automatically set the image tag for articles, but on other pages (like your home page) you'd need to set the image specifically with the following tag:
{JFBCGraph image=http://site.com/link-to-image.jpg}
You can set this on just your home page by creating a Custom HTML module, setting it to only display on the home page menu item, and typing the text above into the text area.
Additionally, you can set a site-wide default image for all pages that don't get one set automatically by JFBConnect and don't have a {JFBCGraph image=xxx} tag. You can set it by typing the following into the "Defaults" box in the Open Graph tab in the Social Configuration area:
image=http://site.com/link-to-default-image.jpg
For more information on Open Graph tags, see the page below:
www.sourcecoast.com/jfbconnect/features/...pen-graph-for-joomla
Hope that helps,
Alex