You have to go to the control screen, the one where you can select which player uses which control, and change the cursor options. To acccess these options, press the shoot button.
You'll see that it says 'Player name' under 'Cursor name'. Cycle through the options until you get 'No' (meaning nothing will be shown) or '<1>'. I don't remember which one of those is the standard one, as I use player names.