Dave McHale

– This plugin has been adopted! Please review the description below. –
A 404 page is what a user sees when they try to reach a non-existent page on your site (because they’ve clicked on a broken link, the page has been deleted or they’ve mistyped a URL).
While the standard 404 page can vary depending on your web host, it doesn’t usually provide the user with much useful information and users may just surf away from your site. Therefore, Google recommend creating a custom 404 page that provides the user with more information about your site and its content.
The Google 404 widget is a quick and easy way to embed a search box on your custom 404 page and provide users with useful information designed to help them find the information they need. Where it can, the Google widget will also suggest other ways for the user to find the information they need, thus increasing the likelihood that they’ll continue to explore your site.
For best results, you should set up a <a href="http://www.google.com/webmasters/" title="Google" rel="nofollow ugc">Google webmaster account</a> and submit an XML sitemap of your blog to ensure the Google 404 widget works effectively.
Find out more <a href="https://support.google.com/webmasters/answer/136085" rel="nofollow ugc">here…</a>
<h4>Features</h4>
Currently the plugin outputs the Google 404 widget in US English by default, or tries to use the current WordPress language. 
You can use the <code>google_404_language</code> filter to specify a different language code.
Also looking to add the possibility to <a href="https://support.google.com/webmasters/answer/100044" rel="nofollow ugc">change the appearance</a> of the widget.
<h4>Please note</h4>
The Google 404 widget plugin is NOT a WordPress widget. 
It is simply the term Google use to describe their enhanced 404 page script.
<h4>Adoption Story</h4>
This plugin was originally developed by <a href="https://profiles.wordpress.org/husobj/" rel="nofollow ugc">Ben Huson</a> and was adopted in September 2014 thanks to the <a href="https://wordpress.org/plugins/tags/adopt-me" rel="ugc">adopt-me</a> tag.
<h4>Coming Soon!</h4>
The ability to customize the CSS from an options page within the WordPress Admin area, attempt to dynamically inject the Google search box without needing to modify your template files, and more!

404 Widget for Google

Add Google search capabilities to your site's 404 "page not found" template.

jfkconsulting

This plugin allows you to take full control over how your WordPress site handles 404 (Not Found) errors. Instead of the default theme’s 404 page, you can:
<ul>
<li>Display a Custom 404 Page: Choose any published page on your site to serve as your 404 error page.</li>
<li>Log 404 Errors: Keep track of which URLs are causing 404 errors, including the referring URL, IP address, and user agent. This helps identify broken links or malicious activity.</li>
<li>Analyze Logs: View statistics like the most frequent 404 URLs and top referrers directly in your WordPress admin.</li>
<li>Create Automatic Redirects: Easily set up 301 (permanent) or 302 (temporary) redirects for specific 404 URLs to send visitors to the correct content. Track redirect hits.</li>
<li>Export Logs: Download your 404 error logs as a CSV file for external analysis.</li>
<li>Optionally Redirect All 404s to Homepage: A simple option to redirect all 404 errors directly to your site’s homepage.</li>
<li>Add <code>noindex</code> Tag: Automatically add a <code>noindex, follow</code> meta tag to your 404 pages (custom or default) to prevent search engines from indexing them.</li>
</ul>

Custom 404 Handler

Customized 404 page, error logging with analysis, automatic redirects and export functionality.

404 Widget for Google vs Custom 404 Handler