Include the 404 — problem solvers will appreciate it
There’s no better or more concise way to communicate the nature of the problem to experts who may have to help your user.
“404” on your error pages is not harmful
Just to get this out of the way, including the error code is not bad, confusing, or annoying. If it were, major web companies like Google, Wikipedia, and the UX experts of Nielsen Norman Group, would not still include them.
Will the error code help your support team?
Any error code is meant to convey specific meaning to people who know its meaning.
It is good UX to help your end users get the help they need as quickly as possible. You should make it as easy as possible for your internal users on the support team to provide that support.
If the people who would provide help to your users would benefit from knowing the HTTP error code, then absolutely include it. You want to help your internal users provide the best help to your external users.
But even if your support team doesn’t need the code (for example, if your site throws serial errors that the support team can look up) there are other good reasons you may want to include standard HTTP error codes:
Consider other experts who might help your user
Knowledge of common HTTP error codes is not limited to web professionals. Some of your users will know what a 404 is, and what might help when you encounter one. If they don’t, they may ask someone in their household, office, etc., who does.
Speaking from personal experience, that nearby expert has often been me. I find it highly annoying when a website displays a “too friendly” error page with no useful information. It’s a little better when a site paraphrases the error so that I can wager a guess about what error was thrown. But even when it’s clear this is a 404 page, not seeing the “404” slows me down a little.
Note: Being a web developer, I can find HTTP errors with the browser tools, if the person I am helping seems really desperate. My internet denizen kids, on the other hand, understand 404’s and some other HTTP error codes (or at least have a gut feel about what to do about them) but don’t know how to use browser tools.
Knowing the error code will help a subset of your users. Why withhold these 3 little information-rich digits?