ORDER BY RAND()

For selecting a single random row with equal distribution and no need to worry about holes, the following might be simpler:

    SET @var = (SELECT FLOOR(RAND() * COUNT(id)) from random);
    PREPARE statement FROM 'SELECT name FROM random LIMIT ?, 1';
    EXECUTE statement USING @var;

I don't know how well this compares in performance to the examples in the article, but it's certainly easier to comprehend.