One of the following will do it. Note that it inserts a row under the row
that is double clicked.
Private Sub Worksheet_BeforeDoubleClick _
(ByVal Target As Range, Cancel As Boolean)
Cancel = True 'Cancel Edit mode started by double click
ActiveCell.Offset(1, 0).EntireRow.Insert Shift:=xlDown
End Sub
The following is similar but will only run if you double click column A.
Private Sub Worksheet_BeforeDoubleClick _
(ByVal Target As Range, Cancel As Boolean)
Cancel = True 'Cancel Edit mode started by double click
If Target.Column = 1 Then
ActiveCell.Offset(1, 0).EntireRow.Insert Shift:=xlDown
End If
End Sub