Hi,
Thank you for the question.
I am assuming that you are referring to Sitewide activity stream and not the stream on user’s profile.

This is doable but there is no good way to achieve it.

I am sharing a proof of concept that will work but I highly discourage using it.

/**
 * Make BuddyPress activity stream random.
 *
 * @author BuddyDev
 * This is a proof of concept that it is doable. We do not endorse doing it as it is error prone.
 */
add_filter( 'bp_activity_paged_activities_sql', function ( $sql, $args ) {
	// Let us find the limit clause.
	$limits = explode( 'LIMIT', $sql );

	// let us break the sql into two parts, one before order by and one after order by.
	$query = explode( 'ORDER BY', $sql );

	$sql = $query[0] . ' ORDER BY RAND() ';
	if ( count( $limits ) == 2 ) {
		$sql .= 'LIMIT ' . $limits[1];
	}

	return $sql;
}, 10, 2 );

Please put this code in your bp-custom.php and it will make sitewide BuddyPress activity random.

Please let me know if it works for you or not?

Regards
Brajesh

You are welcome.
It’s a pleasure to be of service!

Regards
Brajesh