The grid has very extensible OLE D&D support. So, you have several way to approach your problem.
The simplest way to add a custom data object to the D&D operation is to handle the QueryOleDataSourceData event. Here, an event arg will pass the DataObject and you can just add your custom format and daat in this event. Below is a little snippet that puts a specific string in the data object,
private void gridModel_QueryOleDataSourceData(object sender, GridQueryOleDataSourceDataEventArgs e)
{
e.DataObject.SetData(DataFormats.Text, "aaaaaaa");
e.Handled = true;
e.Result = true;
}