My mother-in-law from Spain entered the US under the ESTA/VWP and, by mistake, overstayed her visit by one day. She entered on January 13th and left on April 13th. Today, as she was going through US immigration to re-enter, 4 months after her last entry, she was taken to the Immigration offices, interviewed, and refused entry because of that one day overstay. She will no longer be able to use ESTA. Will this refused entry affect her ability to get a visa?
Yes, it will affect her ability to get a visa. From now on, she has to answer if she was ever refused entry with "yes" and that will lead to extra scrutiny.
It probably won't lead to an automatic rejection of future visa applications, but it will matter. And regardless of the rights or wrongs of the old refusal, she will have to mention it.