Extract value of attribute node via XPath

//Parent[@id='1']/Children/child/@name 

Your original child[@name] means an element child which has an attribute name. You want child/@name.

To get just the value (without attribute names), use string():

string(//Parent[@id='1']/Children/child/@name)

The fn:string() fucntion will return the value of its argument as xs:string. In case its argument is an attribute, it will therefore return the attribute's value as xs:string.