Never actually had a need for this but you will need to use the WP_Comment_Query function. There is this handy comments query generator online to customize this query easily.
But wait, it doesn’t allow you to select a random order. No worries just add or replace the orderby parameter:
'orderby' => 'rand'
This should return comments ordered randomly. Let me know if you need any extra tips.
Edit: the above will not work.
I should have investigated deeper into the subject. The rand argument is not accepted in a comment query, as I thought it would, it is only accepted in post queries.
So how can you achieve this behavior. You can do it in an even simpler way using a built in PHP function to randomize arrays.
You should edit the theme’s comment template, usually it is comments.php and replace the wp_list_comments() function with something like:
//optional arguments in this array, without arguments will return all comments
$args = array();
$comments = get_comments( $args );
// use shuffle to randomize the order of the comments
shuffle($comments);
Now you need to iterate the $comments array, and do something like:
foreach ( $comments as $comment ) :
echo $comment->comment_author . '<br />' . $comment->comment_content;
endforeach;
You can find more info on the get_comments() documentation page and the accepted arguments to customize the query.