Here is sample code for adding tooltip to DropDownList. The key here is adding "title" attribute to the list items in DataBound event.

<asp:DropDownList ID="DropDownList1" runat="server" ondatabound="DropDownList1_DataBound">
</asp:DropDownList>

protected void DropDownList1_DataBound(object sender, EventArgs e)
{
    DropDownList ddl = sender as DropDownList;
    if (ddl != null)
    {
        foreach (ListItem li in ddl.Items)
        {
           li.Attributes["title"] = li.Text; // setting text value of item as tooltip

        }
    }

}