This version may be better.

Checks for blank cells.

Private Sub Worksheet_Change(ByVal Target As Range)
On Error GoTo ws_exit:
If Intersect(Range(Target(1).Address), _
       Range("B:B, F:F, I:I, L:L")) Is Nothing Then Exit Sub
Application.EnableEvents = False
If Target.Value <> "" Then
Target.Offset(0, 1).Value = Now()
End If
ws_exit:
    Application.EnableEvents = True
End Sub

Gord

Through what?

The event code for a non-shared worksheet?

NOTE:  will not work on a Shared Workbook as I said earlier.

1.   Select column A and any other range you want unlocked.
2.   Format>Cells>Protection>  uncheck the "locked" box.
3.   Widen column A to about 16 minimum.

Select the sheet tab and "View Code".   Copy/paste this code into that sheet
module.

Private Sub Worksheet_BeforeDoubleClick(ByVal Target _
As Range, Cancel As Boolean)
On Error GoTo enditall
Application.EnableEvents = False
If Target.Cells.Column = 1 Then
ActiveSheet.Unprotect Password:="justme"
   For Each cell In Target
   If cell.Value = "" Then
   With cell
        .Value = Now
        .Locked = True
   End With
   End If
Next
Cancel = True
End If
enditall:
Application.EnableEvents = True
ActiveSheet.Protect Password:="justme"
End Sub

Double-click on any blank cell in column A and get the date/time entered and
cell locked so timestamp cannot be changed.

Does not affect protection on other unlocked cells.

"justme" can be any password you choose.

You will of course want to lock the code from prying eyes so..........

Alt + F11 to open Visual Basic Editor.

Select your workbook/project and right-click>VBAProject
Properties>Protection>Lock for Viewing.  Apply a password and save/close the
workbook.

When re-opened, code will be unviewable or uneditable.

Gord