And then you'd just take the first result, right? Because it's random?
Due to the LIMIT clause, there is only one row in the result set.
An alternative method which may be better in terms of pseudo-random number generation and performance would be:
$count = mysql_result(mysql_query("SELECT COUNT(*) FROM tablename"), 0);
$num = mt_rand(0, $count - 1);
$result = mysql_query("SELECT columns FROM tablename LIMIT $num, 1");
The idea here being to find out the number of rows in the table, generate a pseudo-random number in the range of the number of rows, and then use that to directly select the row required.