It is not unreasonable to suppose that lap desks were influence by writing boxes made in such countries as Japan, Korea, and China. Apparently, "boxes with compartments for an ink stone, ink stick, water dropper, and brushes, became popular during the Muromachi period (1392-1573)." They were often highly elaborate decorated with lacquer textures and finishes. Here is the picture of a very simple one that appeals to me.
I was tempted at the last antique show in Brimfield to buy a much more elaborate one in metal. I resisted the temptation and am glad, as I need one of these implements as much as I need a hole in my head.