Try this
In Common Page ASPX:
Write a javascript function to get the Iframe ID and store it in a hidden field as
function SetWidgetName() {
document.getElementById('<%= hdnWidgetName.ID%>').value = window.frameElement.id
}
<asp:Button ID="Button1" OnClientClick="SetWidgetName();" runat="server" OnClick="Button1_Click" Text="TestButton"></asp:Button>
<asp:HiddenField ID="hdnWidgetName" runat="server" />
Code behind:
protected void Button1_Click(object sender, EventArgs e)
{
string widgetName = hdnWidgetName.Value;
}