catch the click event on a specific mesh in the renderer

You can generate a callback like this. First define your callback function for each object:

mesh.callback = function() { console.log( this.name ); }

Then follow the standard picking pattern:

var raycaster = new THREE.Raycaster();
var mouse = new THREE.Vector2();

function onDocumentMouseDown( event ) {

    event.preventDefault();

    mouse.x = ( event.clientX / renderer.domElement.clientWidth ) * 2 - 1;
    mouse.y = - ( event.clientY / renderer.domElement.clientHeight ) * 2 + 1;

    raycaster.setFromCamera( mouse, camera );

    var intersects = raycaster.intersectObjects( objects ); 

    if ( intersects.length > 0 ) {

        intersects[0].object.callback();

    }

}

EDIT: updated to three.js r.70

1. Create a click handler

   window.addEventListener('click', onDocumentMouseDown, false);
   

2. Define the function onDocumentMouseDown, note that raycaster the difference in above answer is the index position of the object clicked!

   var raycaster = new THREE.Raycaster();
   var mouse = new THREE.Vector2();
   function onDocumentMouseDown( event ) {
   event.preventDefault();
   mouse.x = ( event.clientX / renderer.domElement.clientWidth ) * 2 - 1;
   mouse.y = - ( event.clientY / renderer.domElement.clientHeight ) * 2 + 1;
   raycaster.setFromCamera( mouse, camera );
   console.log(scene.children);
   var intersects = raycaster.intersectObjects( scene.children );
   console.log(intersects[1]);
   if ( intersects.length > 0 ) {
       intersects[1].object.callback();
   }}
   

3. Define the Mesh object

   var mesh_menu_title = new THREE.Mesh(geometry_menu, materials_menu);
   mesh_menu_title.name = 'select_lang';
   mesh_menu_title.callback = function() { select_language();}
   scene.add(mesh_menu_title);
   

4. define the callback function

   function select_language(){
   var selectedObject = scene.getObjectByName("select_lang"); 
   scene.remove( selectedObject );
   var selectedObject = scene.getObjectByName("start");
   scene.remove( selectedObject );
   var selectedObject = scene.getObjectByName("menu");
   scene.remove( selectedObject );
   }
   

So this code above will handle specific object clicked inside my canvas, then callback a function, the "mesh.callback" and it will remove some scene childs from the canvas.

It doesn't work if you use intersects[0].object.callback(); because at the index 0 the stored object are the vertices.