You can use wp_get_post_categories to get the categories from your posts. Then you can use array_rand to get one random category from that array.
Check out these links on how to use wp_get_post_categories and array_rand to suit your needs:
https://developer.wordpress.org/reference/functions/wp_get_post_categories/
https://www.php.net/manual/en/function.array-rand.php