Why do you want an Items property? You already have the Item property and you add things to the list by using myList.Add("abc"). The Item(index) property is the actual string and you can't subclass the string data type. If you want your collection to have other methods then just create a new subclass.
Code:
Public Class MyCollection
  Inherits List(Of String)

  Public Sub Xy(ByVal index As Integer)
    'do whatever
  End Sub
End Class
Now you can do this:
Code:
Dim myList As New MyCollection
myList.Add("abc")
myList.Xy(index)