I wasn't able to get this working within views proper, so I added a subquery instead.
There are three relevant things that need to be set up in your initial view to get this to work:
- Filter Criteria: Content: Type = 'my content type' (blog post in
the attached image)
- Sort Criteria: Global: Random
- Pager: Use Pager: Display a specified number of items: 1

As it is, that will get you a view that will pull in 1 random result from among all content of type "my content type".
But the question is asking you to pull in 1 item from the last 5 created posts, not from all posts.
To do that, you need to add an additional where condition. That can be accomplished with this snippet:
/**
* Implements hook_views_query_alter().
*
* Limits results to the most 3 recent blog posts.
*/
function MY_MODULE_views_query_alter(&$view, &$query) {
if ($view->name == 'MY_VIEW_NAME' && $view->current_display == 'MY_CURRENT_DISPLAY') {
$query->add_where_expression('AND', "node.created > (SELECT node.created AS created
FROM
node node
WHERE (type = 'MY_CONTENT_TYPE')
ORDER BY created DESC
LIMIT 1 OFFSET 5)");
}
}
That snippet says:
"Find me the creation timestamp of the node that is the fifth-newest. Then filter my final result by nodes with creation timestamps greater than that."
There might be a better way, but this works for me.