When I was in elementary school, we usually called cartridges, "tapes". "Level" and "stage" were common terms, but we also used "board" with them interchangeably.
They seemed a little odd even at the time, but I guess they actually have legitimate origins. Many old computer games actually WERE stored on cassette tapes, so it's easy to see how someone could have gotten used to calling all game media "tapes" and continued to use the term even after they changed formats. And "board" makes more sense in terms of old single-screen arcade games. You could easily say you "cleared the board" when you killed every enemy on a Space Invaders screen, for example. I think it might go back to actual board games, like chess possibly.
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