this should help you debug the issue
$('#rename').click(function () {
    //alert( $(this).closest('td').prev().prev().text());
    var name = prompt('enter the new name');
    if (name) $.ajax({
        url: url() + '/rename',
        type: 'POST',
        data: { value: name,
            id: $(this).closest('tr').find('td :first').attr('value')
        },
        success: function (result) {
            alert('success');
            if (result.content == 1) {//location.reload();
                alert(name);
                //$(this).closest('td').prev().prev().text(name);
            }
        },
        error: function () {
            alert('error');
        }
    })
});
basically added a couple of extra points.
- How do you know success is happening and result.context just isn't equal to 1, so added the alert('success');
 
- 
How do you know it's resturning success at all? Added
error: function () { alert('error'); }
 
- 
you were missing a ; at the end. :)