First of all, I think you want
<?php
mysql_select_db(name of database);
$quotes = "SELECT author,quote FROM inspirational_quotes ORDER BY RAND() LIMIT 1";
$result = mysql_query($quotes);
WHILE ($row = mysql_fetch_array($result)):
ENDWHILE;
echo "$result";
?>
but I have an additional suggestion
Preload all the quote IDs
CREATE TABLE quoteID
(
ndx int not null auto_increment,
id int not null,
PRIMARY KEY (ndx)
);
INSERT INTO quoteID (id) SELECT id FROM inspirational_quotes;
Now choose based on the id from quoteID table
SELECT B.author,B.quote FROM quoteID A INNER JOIN inspirational_quotes B
USING (id) WHERE A.ndx = (SELECT CEILING(MAX(ndx) * RAND()) FROM quoteID);
This should scale just fine because the return value for @rnd_id comes from a list of ids with no gaps in the quoteID table.
<?php
mysql_select_db(name of database);
$quotes = "SELECT B.author,B.quote FROM quoteID A INNER JOIN "
. "inspirational_quotes B USING (id) "
. "WHERE A.ndx = (SELECT CEILING(MAX(ndx) * RAND()) FROM quoteID)";
$result = mysql_query($quotes);
$row = mysql_fetch_array($result);
echo "$result";
?>
Give it a Try !!!