As instructed, I put an image 300x225 called screenshot.png in the top level of my (child) theme folder and it displayed nicely on the Appearance Themes wp-admin page.
But when I uploaded a new image (300x225 with same name) and overwrote the old image, it continued to display the old image! It must be a cacheing issue.
To resolve it, remove the old screenshot.png completely, go into Appearance Themes and refresh the page in your browser. This forces WordPress to clear out the old cached image. Then upload your new image, and refresh Appearance Themes again. There is your new image, nicely displayed next to your theme information!