explicit conversion from XElement to string,

You can use the fact that there's an explicit conversion from XElement to string, which returns null for a null XElement reference. You can then use the null-coalescing operator to go from null to an empty string:

var stds = from std in doc.Descendants("student")
           select new
           {
               ID = std.Element("id").Value,
               Name = (string) std.Element("name") ?? "";
           };