use referrer to populate search query

I have wondered for some time why wordpress 404’s aren’t more helpful, especially when it comes to coming to the site from outside, more-so if it’s from google(/search engines).

Essentially, what I’d like to achieve is that if someone follows a link from google and hits a 404 on my site, that the 404 page display the appropriate “not found” message, but then display a set of search results based on the search terms they used in google.

I see many 404 pages including lists to recent posts, popular articles etc, but not usually the search terms themselves.

I’m happy to use a plugin if one exists, but searching for anything with words like 404 and search are bringing up a whole lot of SEO specifics which is not what I’m after (although I think it would have a good effect for SEO)

My thought is the 404 page would just run a new query using the referrer terms (if they exist) and display them in the form of a search result, probably in combination with recent and popular posts and the like.