WIth the classic Weebly drag and drop editor, you can specify a specific image by using a meta tag. Use this URL of an image file with the meta tag like this:
<meta property="og:image" content="https://www.google.com/someimage.jpg" />
Simply replace everything within the double quotes after content= with the URL of your own image, then add it the Header Code field in Settings > SEO in the website editor. You can also do this on a per-page basis by going to the Pages tab, clicking on a page, then adding it to the Header Code field found in SEO Settings. If you haven't already added the image to your site somewhere, you can also add it to a hidden page and then right-click on the image on the hidden page to copy the URL.
For sites built using the Square Online editor, you can do this for every page of your site within the website editor. Navigate to the page, then click on the gear icon on the left and choose Page settings. Go to the Social tab on the popup, then add an image to the Social image option. You can also set one for the whole site by going to Website > SEO & Social Media, click on Update social media, then adding an image on the Social tab of the popup.
Regardless of which editor you have, don't forget to publish afterward. You may also want to use the Facebook debugger to scrape your site again, since Facebook may use older cached images without doing that.
