Make sure the author is a collaborator on the doc! While they probably will be already if they're adding content, if you're copy and pasting a collaborators work, add them to the doc so they can be selected as an author.

Hover over the first line of text (Title) in your page's canvas.

Select "Options."

Select "Show author."

The space below your title and/or subtitle will show a drop-down menu with 'None' as the default selection.

Click the drop-down menu to select from doc collaborators to indicate the author. You may select multiple authors if desired.