Updating a JSON object using Javascript

A plain JavaScript solution, assuming jsonObj already contains JSON:

Loop over it looking for the matching Id, set the corresponding Username, and break from the loop after the matched item has been modified:

for (var i = 0; i < jsonObj.length; i++) {
  if (jsonObj[i].Id === 3) {
    jsonObj[i].Username = "Thomas";
    break;
  }
}

Here it is on jsFiddle.

Here's the same thing wrapped in a function:

function setUsername(id, newUsername) {
  for (var i = 0; i < jsonObj.length; i++) {
    if (jsonObj[i].Id === id) {
      jsonObj[i].Username = newUsername;
      return;
    }
  }
}

// Call as
setUsername(3, "Thomas");

simply iterate over the list then check the properties of each object.

for (var i = 0; i < jsonObj.length; ++i) {
    if (jsonObj[i]['Id'] === '3') {
        jsonObj[i]['Username'] = 'Thomas';
    }
}

$(document).ready(function(){        
    var jsonObj = [{'Id':'1','Username':'Ray','FatherName':'Thompson'},  
               {'Id':'2','Username':'Steve','FatherName':'Johnson'},
               {'Id':'3','Username':'Albert','FatherName':'Einstein'}];

    $.each(jsonObj,function(i,v){       
      if (v.Id == 3) {
        v.Username = "Thomas";
        return false;
      }
    });

alert("New Username: " + jsonObj[2].Username);

});