How can I bind a collection of C# 7.0 tuple type values to a System.Windows.Forms.Listbox and set the display member to one of the elements?

Unfortunately C#7 value tuples cannot be used for data binding because they use fields, while Windows Forms standard data binding works only with properties.

Ivan's answer, definitely describes the case. As a workaround you can use Format event of ListBox to show name filed:

private void listBox1_Format(object sender, ListControlConvertEventArgs e)
{
    e.Value = (((string name, int ID))e.ListItem).name;
}