Make macro work with upper case and lower case

Use UCase (or LCase) on both the value you want to check and the value you want to check against.

Private Sub Worksheet_Change(ByVal Target As Range)

    On Error GoTo Handler
    
    If Target.CountLarge > 1 Then Exit Sub
    
    If Target.Column = 3 And UCase(Target.Value) = UCase("ARRIVED") Then
        Application.EnableEvents = False
        Target.Offset(0, 1) = Format(Now(), "hh:mm")
        Target.Offset(0, 13) = Format(Now(), "mm-dd-yyyy hh:mm")
        Application.EnableEvents = True
    End If

    If Target.Column = 3 And UCase(Target.Value) = UCase("(Absent)") Then
        Application.EnableEvents = False
        Target.Offset(0, 2) = "(Absent)"
        Target.Offset(0, 13) = "(Absent)"
        Application.EnableEvents = True
    End If

    If Target.Column = 5 And UCase(Target.Value) = UCase("DEPARTED") Then
        Application.EnableEvents = False
        Target.Offset(0, 1) = Format(Now(), "hh:mm")
        Target.Offset(0, 12) = Format(Now(), "mm-dd-yyyy hh:mm")
        Application.EnableEvents = True
    End If

Handler:
    Application.EnableEvents = True

End Sub