Frozen date stamp

Your macro should work for you because it enters the actual date and not the formula. If you enter the formula, then the date will change as you described. Do you want the time stamp to be added automatically when someone enters their name in column A?

Copy and paste this macro into the worksheet code module. Do the following: right click the tab name for your sheet and click 'View Code'. Paste the macro into the empty code window that opens up. Close the code window to return to your sheet. Enter a name in column A and exit the cell.

Private Sub Worksheet_Change(ByVal Target As Range)
    If Intersect(Target, Range("A:A")) Is Nothing Then Exit Sub
    Target.Offset(0, 1) = Date
End Sub

Did you place the macro in the code module for the worksheet and not in a standard module?

The file you posted works for me. Don't forget to exit the cell or press the RETURN key after you enter the name. This macro is a Worksheet_Change event which means that it is triggered automatically when you change the value of a cell. In your case, the change is restricted to column A. This type of macro must go into the code module for the worksheet (which you did correctly) and not in a standard module.