これを試して

HTML:

   <div ng-app="myApp">
    <section class="vbox" ng-controller="WebsiteController">
   <section class="panel animated fadeInDown">
<header class="panel-heading">
  <h3>{{mySelectedUser.last_name}} </small> <!-- Just to test I print it here -->
</header>
<div class="panel-body m-b">
    <div class="table-responsive">
        <table class="table table-striped b-t text-sm">
          <thead>
              <tr>
                <th>URL</th>
                <th>Assigned to</th>
                <th>Action</th>
              </tr>
            </thead>
            <tbody>
                <tr ng-repeat="website in websites">
                    <td>
                         <span ng-repeat="assigned in website.users"> <span class="label label-info"> {{assigned.first_name}} </span> &nbsp; </span>
                         <a ng-click="addUser()" ng-hide="adding" class="btn btn-success btn-xs"> Add new </a></span>                               
                             <select ng-hide="!adding" 
                                        name="myselect" 
                                        ng-model="selectedUser" 
                                        ng-change="abc(selectedUser)"
                                        ng-options="u.first_name for u in users">
                             </select>
                         <a href="#" ng-hide="!adding" ng-click="cancelAddUser()" class="btn btn-warning btn-xs">Cancel</a>
                         <input type="text" ng-model="selectedUser.first_name"> <!-- It works here! -->
                    </td>
                    <td>
                        <a href="#" class="btn btn-xs btn-white"> <i class="icon-pencil"></i> Edit </a>
                    </td>
                </tr>
            </tbody>                    
        </table>
    </div>
</div>
 </section>
</section>
</div>    

コントローラー：

var JApp = angular.module('myApp', []);

 JApp.controller('WebsiteController', function($scope) {
//var UsersService = $resource('/auth/users', {});

$scope.adding = false;
$scope.users = [];

$scope.websites = [
    {
        "id": 1,
        "url": "www.google.com",
        "cms": "Wordpress"
    }
    ];

$scope.addUser = function() {
    $scope.adding = true;
   var data = [
   {
     "id": 1,
     "role_id": 1,
     "created_at": "2013-09-19 05:54:36",
     "updated_at": "2013-09-19 05:54:36",
     "email": "admin@admin.com",
     "username": "admin",
     "first_name": "Admin",
     "last_name": "Istrator"
   },
  {
     "id": 2,
     "role_id": 2,
     "created_at": "2013-09-19 05:54:36",
     "updated_at": "2013-09-19 05:54:36",
     "email": "johndoe@gmail.com",
     "username": "john",
     "first_name": "John",
     "last_name": "Doe"
  }
];
   // UsersService.query({}, function(data) {
        $scope.users = data; // suppose data is coming from UsersService.query
        $scope.selectedUser = data[1];
        $scope.mySelectedUser = $scope.selectedUser ;
  //  });
}
$scope.abc = function(a) {
    $scope.mySelectedUser = a;

 }
$scope.cancelAddUser = function() {
    $scope.adding = false;
}
});

デモを見る

選択ドロップダウン オプションの値は u.first_name であるため、次のいずれかを行います。

$scope.selectedUser = data[0].first_name;

または、ID などを使用するように変更します。

ng-options="u.id as u.first_name for u in users"
$scope.selectedUser = data[0].id;