After overcoming frustration about WebView.saveWebArchive() format change in Android 4.4, I tried the "unknown google project" Chitranshu Asthana mentioned in his answer, but code provided there is slow (~10s for 1MB *.mht file with a dozen of pictures) and doesn't handle attached file names correctly.
MHT Unpack library combined with Java Mail for Android (not the one provided by Oracle) worked perfectly for me.
EDIT: Fixed the link to MHT Unpack library. Also, here's usage example:
// contentPath - path to input .mht file
public static String unpackMht(String contentPath) throws IOException {
        // dstPath - path where file will be unpacked
        String dstPath = openTempDir(null) + File.separator;
        String indexFileName = dstPath + new File(contentPath).getName();
        try {
            Collection<Attachment> attachments = MHTUnpack.unpack(new File(contentPath));
            for (Attachment attachment : attachments) {
                String filename = attachment.getFileName();
                String path = filename == null ? indexFileName : dstPath +  filename;
                File newFile = new File(path);
                if (newFile.exists()) {
                    newFile.delete();
                }
                attachment.saveFile(path);
            }
            return indexFileName;
        } catch (MessagingException e) {
            throw new IOException(e);
        }
    }