OK I woudl add a checkbox to the Field_Edit.ascx file

<asp:CheckBox ID="DeleteCheckBox" runat="server" test="Delete" />

 then in the ExtractValues method check the checkbox if set set the dictionary to null this will null the value

protected override void ExtractValues(IOrderedDictionary dictionary)
{
    if (FileUpload1.HasFile)
    {
        string fileAttributeFullPath = Utility.FrontendPhysicalPath + GetFileAttributePath();
        string fileName = Utility.GetRand() + "-" + FileUpload1.FileName;

        FileUpload1.SaveAs(fileAttributeFullPath + fileName);
        if (!String.IsNullOrEmpty(ViewState["FieldValueString"].ToString()))
            try
            {
                File.Delete(fileAttributeFullPath + ViewState["FieldValueString"].ToString());
            }
            catch { }

        dictionary[Column.Name] = fileName;
    }
    if (DeleteCheckBox.Checked)
        dictionary[Column.Name] = null;
}

give that a try.