You would have to declare a method outside of Thing (or a static Thing method), then you could pass a method-group reference to it:
private string GetName(Thing thing)
{
return thing.Name;
}
...
List<String> nameList1 = thingList.Select(GetName).ToList();
In C# 6, you can also use an expression-bodied function to save a couple of lines:
private string GetName(Thing thing) => thing.Name;