Client Website always return Null Json String

The error seems to be here:

var jsonObject = '{ "String": ' + JSON.stringify(activityMap) + '}';

Remove that line, as you do not need it and change your AJAX call to the following:

$.ajax({
        type: 'POST',
        url: 'http://localhost:52535/PUendeligService.svc/AddNewActivity',
        data: {jsonObject: JSON.stringify(activityMap) },
        contentType: 'application/json; charset=utf-8',
        dataType: 'json',
        processData: true,
        success: function (data, status, jqXHR) {
            alert("Success: " + data);
        },
        error: function (xhr) {
            console.log(xhr.responseText);
            alert("Error: " + xhr.responseText);
        }
    });

The problem is that your service expects "jsonObject" variable, but you are sending "String". Hope this helps.

Let's simplify your sample a bit and start with this

<script type="text/javascript">
    function OnModalCreateNewActivityBtnClick() {

        var data = {
            Status: modal.find('#new-activity-modal-status-dropdown').val(),
            Name: modal.find('#new-activity-modal-name-field').val()
        };

        $.ajax({
            type: 'POST',
            url: 'Service.svc/AddNewActivity',
            data: JSON.stringify(data), 
            contentType: 'application/json; charset=utf-8', 
            dataType: 'json', 
            success: function (msg) {
                alert(data.Status + ' == ' + msg.Status);
            },
            error: function (e) {
                alert('failed with ' + e.statusText);
            }
        });
    }
</script>

Your activity class

   [DataContract]
    public class Activity
    {
        [DataMember]
        public String Status
        {
            get; set;
        }

        [DataMember]
        public String Name
        {
            get; set;
        }
    }

Implementation

[WebInvoke(Method ="POST", RequestFormat = WebMessageFormat.Json, ResponseFormat = WebMessageFormat.Json, BodyStyle = WebMessageBodyStyle.Bare, UriTemplate = "AddNewActivity/")]
public Activity AddNewActivity(Activity activity)
{
  // your stuff
  return activity;
}

Lastly, since you didn't show your configuration let's assume the following

     <system.serviceModel>
    <services>
      <service name="PUendelig.Service"  behaviorConfiguration="serviceBehavior">
        <endpoint address="" binding="webHttpBinding" contract="PUendelig.IService" behaviorConfiguration="web"/>
      </service>
    </services>
    <behaviors>
      <serviceBehaviors>
        <behavior name="serviceBehavior">
          <serviceMetadata httpGetEnabled="true" httpsGetEnabled="true"/>
          <serviceDebug includeExceptionDetailInFaults="false"/>
        </behavior>
      </serviceBehaviors>

      <endpointBehaviors>
        <behavior name="web">
          <webHttp/>
        </behavior>
      </endpointBehaviors>

    </behaviors>

    <protocolMapping>
      <add binding="basicHttpsBinding" scheme="https"/>
    </protocolMapping>
    <serviceHostingEnvironment aspNetCompatibilityEnabled="true" multipleSiteBindingsEnabled="true"/>
  </system.serviceModel>

I put this together in github for you to download. Please feel free to adjust as need be and help others in the process https://github.com/alexnolasco/SO35094908