As seen on WP Mix: http://wp-mix.com/wordpress-404-email-alerts/

Here is a customized plug-n-play PHP script for reporting 404 errors via email. No editing required, just clean email alerts for each 404 error.

To implement, simply include this script at the top of your theme’s 404.php file:

// WP 404 ALERTS @ http://wp-mix.com/wordpress-404-email-alerts/

// set status
header("HTTP/1.1 404 Not Found");
header("Status: 404 Not Found");

// site info
$blog = get_bloginfo('name');
$site = get_bloginfo('url') . '/';
$email = get_bloginfo('admin_email');

// theme info
if (!empty($_COOKIE["nkthemeswitch" . COOKIEHASH])) {
 $theme = clean($_COOKIE["nkthemeswitch" . COOKIEHASH]);
} else {
 $theme_data = wp_get_theme();
 $theme = clean($theme_data->Name);
}

// referrer
if (isset($_SERVER['HTTP_REFERER'])) {
 $referer = clean($_SERVER['HTTP_REFERER']);
} else {
 $referer = "undefined";
}
// request URI
if (isset($_SERVER['REQUEST_URI']) && isset($_SERVER["HTTP_HOST"])) {
 $request = clean('http://' . $_SERVER["HTTP_HOST"] . $_SERVER["REQUEST_URI"]);
} else {
 $request = "undefined";
}
// query string
if (isset($_SERVER['QUERY_STRING'])) {
 $string = clean($_SERVER['QUERY_STRING']);
} else {
 $string = "undefined";
}
// IP address
if (isset($_SERVER['REMOTE_ADDR'])) {
 $address = clean($_SERVER['REMOTE_ADDR']);
} else {
 $address = "undefined";
}
// user agent
if (isset($_SERVER['HTTP_USER_AGENT'])) {
 $agent = clean($_SERVER['HTTP_USER_AGENT']);
} else {
 $agent = "undefined";
}
// identity
if (isset($_SERVER['REMOTE_IDENT'])) {
 $remote = clean($_SERVER['REMOTE_IDENT']);
} else {
 $remote = "undefined";
}
// log time
$time = clean(date("F jS Y, h:ia", time()));

// sanitize
function clean($string) {
 $string = rtrim($string);
 $string = ltrim($string);
 $string = htmlentities($string, ENT_QUOTES);
 $string = str_replace("n", "<br>", $string);

if (get_magic_quotes_gpc()) {
 $string = stripslashes($string);
 }
 return $string;
}

$message =
 "TIME: " . $time . "n" .
 "*404: " . $request . "n" .
 "SITE: " . $site . "n" .
 "THEME: " . $theme . "n" .
 "REFERRER: " . $referer . "n" .
 "QUERY STRING: " . $string . "n" .
 "REMOTE ADDRESS: " . $address . "n" .
 "REMOTE IDENTITY: " . $remote . "n" .
 "USER AGENT: " . $agent . "nnn";

mail($email, "404 Alert: " . $blog . " [" . $theme . "]", $message, "From: $email");

No editing required, just include and done. To disable the email reporting, either remove the script or comment-out the mail() function, like so:

// mail($email, "404 Alert: " . $blog . " [" . $theme . "]", $message, "From: $email");

Note that any of the variables (e.g., $time, $request, et al) are available for use elsewhere on the page. For example, you could display the information to the visitor to help deliver a useful 404 page.

Here is a screenshot showing how the email alerts will look in your inbox.

Update! Thanks to Drew Jaynes for sharing his OOP-enhanced version of the 404-email script. Muy bueno!!