Renombrar filas duplicadas

With Dups As
    (
    Select Id, Name
        , Row_Number() Over ( Partition By Name Order By Id ) As Rnk
    From Table
    )
Select D.Id
    , D.Name + Case
                When D.Rnk > 1 Then ' (' + Cast(D.Rnk As varchar(10)) + ')'
                Else ''
                End As Name
From Dups As D

Si desea una declaración de actualización, puede usar prácticamente la misma estructura:

With Dups As
    (
    Select Id, Name
        , Row_Number() Over ( Partition By Name Order By Id ) As Rnk
    From Table
    )
Update Table
Set Name = T.Name + Case
                    When D.Rnk > 1 Then ' (' + Cast(D.Rnk As varchar(10)) + ')'
                    Else ''
                    End
From Table As T
    Join Dups As D
        On D.Id = T.Id