To sort results randomly within a FacetWP Template, you can set orderby to rand:

<?php
return [
  "post_type" => [
    "post"
  ],
  "post_status" => [
    "publish"
  ],
  "posts_per_page" => "10",
  "orderby" => [
    "rand" => "ASC", // sort by random
    "title" => "ASC" // title as a fallback sort
  ]
];

Since FacetWP is ajax-based, the problem is that the results will be re-randomized on each facet interaction (including pagination), which is less than desirable.

Fortunately, we can fix this issue with some extra code in functions.php:

<?php
function preserve_random_order( $orderby ) {
    $seed = floor( time() / 10800 ); // randomize every 3 hours
    $orderby = str_replace( 'RAND()', "RAND({$seed})", $orderby );
    return $orderby;
}
add_filter( 'posts_orderby', 'preserve_random_order' );

Hat tip to Alex Durston at Amp Web Design for the code example.

See also