Make a query with the return of an Ajax

Your ajax does nothing to complete, however you should do an action of which would fill your list with the elements returned in this way for example:

$.ajax({
    type : 'POST',
    url  : 'index.php/home',
    data : {id : $('#formProd select option:selected').val()}
}).done(function() {
  var len = data.id.length; //assumindo que vai retornar um Array.
  var List = []; //lista de elementos retornados
  for(var i=0; i < len; i++){ //percorre a lista de elementos retornados
    List.push("<li>"+data.id[i]+"</li>"); 
  }
  $('#linkList').html(List.join('')); //coloca as li's no ul com os elementos retornados.
});

On the Query you want to do, you can do something like this:

$id = $_POST['id'];
if (is_numeric($id))
  $sql = 'SELECT * FROM sua_tabela WHERE id='.$id;

Note that I checked first if the return of $_POST['id'] is numero, so that a SQL Injection does not happen in your application.

If the problem is to send the option ID selected via Ajax, the code below should work without problems:

$('#formProd select').change(function(){
    $.ajax({
        type: 'POST',
        url: 'index.php/home',
        data: { id: $(this).val() },
        //contentType: 'application/json; charset=utf-8',
        //dataType: 'json',
        success: function (data) {
            // utilizar o retorno
        }
    });
});

Note that to fill in the value of the ID you want to send you do not need to access the value this way: $('#formProd select option:selected') , since you already accessed the component via the "change" event, just use $(this).val() to access the value of the selected option ID.