Selecting one row from MySQL using mysql_* API

Try with mysql_fetch_assoc .It will returns an associative array of strings that corresponds to the fetched row, or FALSE if there are no more rows. Furthermore, you have to add LIMIT 1 if you really expect single row.

$result = mysql_query("SELECT option_value FROM wp_10_options WHERE option_name='homepage' LIMIT 1");
$row = mysql_fetch_assoc($result);
echo $row['option_value'];

use mysql_fetch_assoc to fetch the result at an associated array instead of mysql_fetch_array which returns a numeric indexed array.

$result = mysql_query("SELECT option_value FROM wp_10_options WHERE option_name='homepage'");
$row = mysql_fetch_assoc($result);
echo $row['option_value'];

Functions mysql_ are not supported any longer and have been removed in PHP 7. You must use mysqli_ instead. However it's not recommended method now. You should consider PDO with better security solutions.

$result = mysqli_query($con, "SELECT option_value FROM wp_10_options WHERE option_name='homepage' LIMIT 1");
$row = mysqli_fetch_assoc($result);
echo $row['option_value'];