I just had the same problem trying to extend jquery UI plugins, and here is the solution I found (found it through jquery.ui.widget.js):

(function($) {
/**
 *  Namespace: the namespace the plugin is located under
 *  pluginName: the name of the plugin
 */
    var extensionMethods = {
        /*
         * retrieve the id of the element
         * this is some context within the existing plugin
         */
        showId: function(){
            return this.element[0].id;
        }
    };

    $.extend(true, $[ Namespace ][ pluginName ].prototype, extensionMethods);


})(jQuery);

hope this helps, please ask if you have any questions.

I had the same issue and came here, then Jared Scott s answer inspired me.

(function($) {

    var fullCalendarOrg = $.fn.fullCalendar;

    $.fn.fullCalendar = function(options) {
        if(typeof options === "object") {
            options = $.extend(true, options, {
                // locale
                isRTL: false,
                firstDay: 1,
                // some more options
            });
        }

        var args = Array.prototype.slice.call(arguments,0);
        return fullCalendarOrg.apply(this, args);
    }

})(jQuery);

Ive found that with a lot of plugins the methods are protected/private (ie in the closures scope). If yo need to modify the functionality of the methods/functions then your out of luck unless youre willing to fork it. Now if you dont need to change any of these methods/functions then you can use $.extend($.fn.pluginName, {/*your methods/properties*/};

Another thing ive ended up doing before is simply using the plugin as a property of my plugin instead of trying to extend it.

What it all really comes down to is how the plugin you want to extend is coded.

$.fn.APluginName=function(param1,param2)
{
  return this.each(function()
    {
      //access element like 
      // var elm=$(this);
    });
}

// sample plugin
$.fn.DoubleWidth=function()
  {
    return this.each(function()
      {
        var _doublWidth=$(this).width() * 2;
        $(this).width(_doubleWidth);
      });
  }

//

<div style="width:200px" id= div! >some text</div>

// using custom plugin

$( #div1 ).DoubleWidth();

/// above written type of utils usually work of dom elements /////////////// custom utils

(function($){
  var _someLocalVar;
  $.Afunction=function(param1,param2) {
    // do something
  }
})(jquery);

// access above util as

$.Afunction();

// this type of utils usually extend javascript

My approach in rewriting jQuery plugins has been to move methods and variables that need to be accessed to the options block and call the extend

// in the plugin js file
$.jCal = function (target, opt) {
    opt = $.extend({
       someFunctionWeWantToExpose: function() {
           //  this  refers to  opt , which is where are our required members can be found
       }
    }

    // do all sorts of things here to initialize

    return opt; // the plugin initialisation returns an extended options object
}


////// elsewhere /////

var calendar = $("#cal1").jCal();
calendar.someFunctionWeWantToExpose();

Example similar to Jared Scott`s answer, but making a copy of original object prototype gives the ability to call parent method:

(function($) {

    var original = $.extend(true, {}, $.cg.combogrid.prototype);

    var extension = {
        _renderHeader: function(ul, colModel) {
            original._renderHeader.apply(this, arguments);

            //do something else here...
        }
    };

    $.extend(true, $.cg.combogrid.prototype, extension);

})(jQuery);

jQuery Widget can be extended using jQuery Widget Factory.

(function ($) {
    "use strict";

    define([
        "jquery",
        "widget"
    ], function ($, widget) {
        $.widget("custom.yourWidget", $.fn.fullCalendar, {
            yourFunction: function () {
                // your code here
            }
        });

        return $.custom.yourWidget;
    });
}(jQuery));

Check out jQuery Documentation to learn more:
Widget Factory API
Extending Widgets with the Widget Factory