Rest hat gefehlt, verzeih.

Folgende Zeile will das aktive Sheet auf strNewName umbenennen (nachdem kopiert wurde)

ActiveSheet.Name = strNewName

Wenn Du auf abbrechen klickst, ist strNewName aber = "".

Und dat geht nicht - versuch folgendes nochmal:

Sub TestA()
Sheets("PRINT").Select
  
Dim strDatum
   Dim Register As Worksheet
   Dim bolShtVorhanden As Boolean
   Dim strNewName As String
   Dim bolErsetzen As Boolean
   Dim vntAntwort As Variant
     
   strDatum = Format(Date, "dd.mm.yyyy")
   ActiveSheet.Copy After:=Sheets(ActiveWorkbook.Sheets.Count)
        
   Do
      strNewName = InputBox("The File will be saved in your current Workbook and today's date: " & Format(Date, "dd.mm.yyyy"), , strDatum)
     
      For Each Register In ActiveWorkbook.Sheets
         If Register.Name = strNewName Then
            bolShtVorhanden = True
            vntAntwort = MsgBox("The file already exists.," & vbCrLf _
               & "Do you want to overwrite it?", _
               vbQuestion + vbYesNo, "Security check")
            If vntAntwort = vbYes Then
               bolErsetzen = True
               Exit Do
            End If
         End If
      Next Register
      If Not bolShtVorhanden Then Exit Do
   Loop
     
   If bolErsetzen Then
      On Error Resume Next
      With Application
         .ScreenUpdating = False
         .DisplayAlerts = False
         Sheets.Delete
         .DisplayAlerts = True
         .ScreenUpdating = True
           
      End With
   End If
  
 
    If Not strNewName = "" Then
      ActiveSheet.Name = strNewName
    End if
     
    Range("A1").Select
         
  
Sheets("Welcome").Select
 MsgBox "Saved!"
  
End Sub