How can I rename files with Grunt, based on the respective file’s parent folder name?

This can be done using the grunt-contrib-copy plugin.

The main thing to note is that you can change the destination programmatically by using a rename function (which takes in the destination and source of each file).

Here is a (somewhat brittle) sample Gruntfile.js that should copy to your desired structure:

module.exports = function(grunt) {
  // Project configuration.
  grunt.initConfig({
    copy: {
      main: {
        files: [
          {
            expand: true, 
            cwd: 'src/modules/', 
            src: ['**/*.js'], 
            dest: 'dev/js/', 
            rename: function(dest, src) {
              // use the source directory to create the file
              // example with your directory structure
              //   dest="dev/js/"
              //   src="https://stackoverflow.com/questions/15271121/module1/js/main.js"
              return dest + src.substring(0, src.indexOf("https://stackoverflow.com/")) + '.js';
            }
          },
          {
            expand: true, 
            cwd: 'src/modules/', 
            src: ['**/*.scss'], 
            dest: 'dev/css/', 
            rename: function(dest, src) {
              return dest + src.substring(0, src.indexOf("https://stackoverflow.com/")) + '.css';
            }
          },
          {
            expand: true, 
            cwd: 'src/modules/', 
            src: ['**/*.html'], 
            dest: 'dev/', 
            rename: function(dest, src) {
              return dest + src.substring(0, src.indexOf("https://stackoverflow.com/")) + '.html';
            }
          }
        ]
      }
    }
  });

  grunt.loadNpmTasks('grunt-contrib-copy');

  // Default task(s).
  grunt.registerTask('default', ['copy']);
};