Either the style you are attempting to rename doesn't exist in the ActiveDocument, or you are trying to rename a built-in style, which isn't possible.
You need to add some error checking, for example:
Code:
Sub RenameCharacterStyles()
Dim oldNames As Variant
oldNames = Array("Old Style 1", "Old Style 2", "Old Style 3")
Dim newNames As Variant
newNames = Array("New Style 1", "New Style 2", "New Style 3")
Dim i As Integer
Dim sty As Style
For i = 0 To UBound(oldNames)
Set sty = ActiveDocument.Styles(oldNames(i))
If Not sty Is Nothing Then
If Not sty.BuiltIn Then sty.Name = newNames(i)
End If
Next i
End Sub