Make sure you have filled out the item page for the story map on ArcGIS Online. The facebook link preview will use the title, subtitle, and thumbnail from the item page.
If you have already updated your item page but you are still seeing old content, you can request Facebook to clear their cache and request the new information with this tool: https://developers.facebook.com/tools/debug/og/object/. Just enter the URL to the story and click the "Fetch new scrape information."
If you are getting the correct title and subtitle, try waiting a few minutes and sharing the link a couple times. Facebook does not cache the photo the first time a link is shared.