My son is in prison in Washington state, and although he has a copy of his discovery file, he can't afford to have the 400 pages copied and mailed to me. He tells me the file is public record. How do I go about getting a copy?

If the matter is of public record, you can either request it from his attorney, who should have a copy, or file a Freedom of Information Act request with the court.

Mail it to you, you copy and mail back or have jurisdiction copy, you will still have to pay for it, unless the attorney has an extra. Will be much cheaper to mail it to you.

I agree, even if you get it from the court, it is going to cost a lot of money for copies.
A couple questions, why do you need it ? There is nothing you can do with it? Why not send him money to his prison account to make the copies

Prisoners often get informal legal advice from "jailhouse lawyers", i.e.: fellow prisoners. Don't necessarily rely upon what he has been told by them.

In this case, it will depend upon who has the file. Most likely, it would be his attorney, in which case it wouldn't be a "public record". A discovery file often is not filed with the clerk of court, so that might not be helpful.

Finally, it would be good to ask him what he thinks he needs it for. If he is working on some sort of post-conviction relief (similar to an appeal), it would be best to get an attorney to do it instead.