How to convert date format in json response to string format in sharepoint?

If you are developing javascript for SharePoint and can use its javascript libraries then it might be a good idea to be using standart approach which take care of all the rest

var date = new Date();
var arg = { date: date };
var str = Sys.Serialization.JavaScriptSerializer.serialize(arg);// "{"date":"\/Date(1377757755979)\/"}"
var newObject = Sys.Serialization.JavaScriptSerializer.deserialize(str);
var newDate = newObject.date; // Date {Thu Aug 29 2013 10:29:15 GMT+0400}

The class is a part of Microsoft Ajax Library so it is a good idea to use it whenever you want to communicate with standart Microsoft Web Services.

        private static DateTime ConvertJsonTimestamp(double jsonTime)
    {

        var baseTime = new DateTime(1970, 1, 1, 0, 0, 0, 0);

        return baseTime.AddSeconds(jsonTime);

    }

pass in your Time variable to this method and you will get a .net date back. There are other methods, but most requires som third party library like Newtonsoft

As part of .Net Framework at least two options are available:

• Deserialize method from System.Web.Script.Serialization.JavaScriptSerializer
• ReadObject method from System.Runtime.Serialization.Json.DataContractJsonSerializer

Example

string dateString = @"""\/Date(1377683175000)\/""";
var serializer = new JavaScriptSerializer();
var date = serializer.Deserialize<DateTime>(dateString);  //returns {28.8.2013 9:46:15}