Background here. I have posts on the book here and here. And for those Aramaic gravestones (there are others), see here.
As for James's question, I have some thoughts.
Mark also quotes Aramaic words attributed to Jesus in 5:41, 7:34, and 15:34 (cf. Matthew 27:46), but in those cases he translates them into Greek.
Matthew does not translate Raka in 5:22, although the next saying does use the near-synonym "fool" (Mōre) in Greek. And it would have been awkward to use the word "fool" twice in the same sentence. Perhaps he positioned the two sayings together to give his Greek readers the sense of the Aramaic word, while keeping the Aramaic, like Mark, for vividness and local color.
But I would not rule out James's idea that Matthew's Greek-speaking readers may have used the Aramaic word raka themselve (as a swear word?). In it's favor is Matthews use of mammon, a (Hebrew/)Aramaic word for "money," untranslated in 6:24 (cf. Luke 16:9, 11, 13). Presumably his readers knew the meaning of the word and maybe they even used it.
Perhaps a closer analogy to James's idea is Paul's use of the Aramaic phrase Marana tha ("our Lord, come!") untranslated in 1 Corinthians 16:22. It looks like Paul and his readers had adopted the Aramaic phrase as a salutation for their own use.
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