private void bindingNavigatorAddNewItem_Click(object sender, EventArgs e)
{
      bindingSource.AddNew();
      bindingSource.EndEdit();
}

            ...
            DataRow newProject = mainForm.handbooksDataSet.Tables["ProjectsTable"].NewRow();
            newProject["name"] = newName;
            mainForm.handbooksDataSet.Tables["ProjectsTable"].Rows.Add(newProject);
            ...

            // определяем максимальный ID
            DataRow[] rows = tempMainForm.handbooksDataSet.Tables["ProjectsTable"].Select();
            DataRow row = rows[rows.Length - 1];
            maxID = (int)row["ID"];

            // настраиваем автоинкремент
            mainForm.handbooksDataSet.Tables["ProjectsTable"].Columns["ID"].AutoIncrement = true;
            mainForm.handbooksDataSet.Tables["ProjectsTable"].Columns["ID"].AutoIncrementSeed = maxID + 1;
            mainForm.handbooksDataSet.Tables["ProjectsTable"].Columns["ID"].AutoIncrementStep = 1;