To update your xml use SetElementValue method of the XElement :
var items = from item in xmlDoc.Descendants("item")
where item.Element("itemID").Value == itemID
select item;
foreach (XElement itemElement in items)
{
itemElement.SetElementValue("name", "Lord of the Rings Figures");
}
EDIT : Yes, I tried your example and it saves updated data to the file. Save your updated xml with Save method of the XDocument, here is the code that I tried :
string xml = @"<items>
<item>
<itemID>1</itemID>
<isGadget>True</isGadget>
<name>Star Wars Figures</name>
<text1>LukeSkywalker</text1>
</item>
</items>";
XDocument xmlDoc = XDocument.Parse(xml);
var items = from item in xmlDoc.Descendants("item")
where item.Element("itemID").Value == "1"
select item;
foreach (XElement itemElement in items)
{
itemElement.SetElementValue("name", "Lord of the Rings Figures");
}
xmlDoc.Save("data.xml");