To show an article in this section, the article must be in the “News Stories” category and marked as “Featured” (the yellow star to the left of the article title in the list view). To order the articles, use the “Search Tools” to select the “News” category (make sure it is the News category that is under News). Then using the sort filter (on the right), choose “Ordering ascending.” You can now drag and drop the articles using the three vertical dots to the left of the article title.
The module (titled "News" and in position-3) should be set to "Ordering" and "Ascending."
The module can be changed if there are articles in the newsfeed that are from other categories (Media Mentions, etc), which may make it harder to organize by "Ordering."
NOTE: There should only be four articles starred/Featured at any time.