AngularJS - extending module with own types / providers

This is a pretty interesting question. I'll prefix my answer with an opinion: I don't think you should extend angular.module to provide a dialog method. Those methods are shortcuts into built-in Angular providers, and the Angular team adds some from time to time. Since you can get access to the functionality you're looking for without adding the dialog method, I wouldn't. That said, the code below does show you how a very basic version of this might work (it doesn't modify the Angular module prototype, just the single instance of the module).

<div ng-app="myApp">
  <div ng-controller='MainController'>
    <div>
      <button ng-click='askName()'>Ask Name</button>
      <button ng-click='askNameAgain()'>Ask Name Again</button>
      <button ng-click='askAge()'>Ask Age</button>
      <button ng-click='askFood()'>Ask Food</button>
    </div>
    <div>{{lastResponse}}</div>
  </div>
</div>

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

// Provide some basic injectables for testing
app.constant('nameString', 'NAME');
app.constant('ageString', 'AGE');
app.constant('foodString', 'FAVORITE FOOD');

// Create the dialog provider
app.provider('dialog', function($provide, $injector) {
  var dialogs = {};

  this.register = function(name, configFn) {
    // Create a new service
    $provide.factory(name, function($window, $q) {
      dialogs[name] = function() {
        // Get data based on DI injected version of configFn
        var data = $injector.invoke(configFn);
        // faking async here since prompt is really synchronous
        var deferred = $q.defer();
        var response = $window.prompt(data.text);
        deferred.resolve(response);
        return deferred.promise;
      };
      return dialogs[name];
    });
  };

  // Injecting the service itself gives you a function that
  // allows you to access a dialog by name, much like $filter
  this.$get = function() {
    return function(name) {
      return dialogs[name];
    };
  };
});

// Providing dialog injectables via app.config
app.config(function(dialogProvider) {
  dialogProvider.register('askFood', function(foodString) {
    return { text: 'What is your ' + foodString + '?' }
  });
});

// Alternatively, shortcut to accessing the dialogProvider via app.dialog
app.dialog = function(name, configFn) {
  app.config(function(dialogProvider) {
    dialogProvider.register(name, configFn);
  });
};

app.dialog('askName', function(nameString) {
  return { text: 'What is your ' + nameString + '?' }
});

app.dialog('askAge', function(ageString) {
  return { text: 'What is your ' + ageString + '?' }
});

app.controller('MainController', 
               function($scope, askName, askAge, askFood, dialog) {
  var setLastResponse = function(result) {
    $scope.lastResponse = result;
  };

  $scope.askName = function() {
    askName().then(setLastResponse);
  };

  $scope.askNameAgain = function() {
    // get the dialog through the dialog service
    // much like how $filter works
    var theDialog = dialog('askName');
    theDialog().then(setLastResponse);
  };

  $scope.askAge = function() {
    askAge().then(setLastResponse);
  };

  $scope.askFood = function() {
    askFood().then(setLastResponse);
  };
});

Here is a working example: http://jsfiddle.net/BinaryMuse/zj4Jq/

By leveraging $injector.invoke inside of your dialogProvider.register function, you can provide the ability to use a key like controller in the data your configFn returns. Since directive works a lot like this already, you may gain a lot from checking out the AngularJS source.