How to modify scope from within a directive in AngularJs

I updated the fiddle, basically had to go to the parent to get the right “selected” variable, also used the isolate scope = to get two way binding between the value passed in and the internal model.

http://jsfiddle.net/nJ7FQ/2/

angular.module('app', [])

    .directive("customTag", [function () {
    return {
        restrict: "E",
        replace: true,
        template: "<input type="button" value="Click me" />",
        scope: {model:'='},

        link: function (scope, element, attrs) {
            element.bind('click', function () {
                scope.model[attrs.selectedItem] = "New value";
                scope.$apply();
            });
        }
    };
}]);

function AppController($scope) {
    $scope.selected = 'Old value';
}

and the HTML

<div ng-app="app" ng-controller="AppController">
    <p>Selected: {{ selected }}</p>
    <div ng-switch on="selected">
        <div ng-switch-default>
            <p>Item: {{ selected }}</p>
            <custom-tag selected-item="selected" model="$parent" />
        </div>
        <div ng-switch-when="New value">
            <p>Worked</p>
        </div>
    </div>
</div>

Updated the fiddle to use your original reading of the property from the attribute:
http://jsfiddle.net/nJ7FQ/4/