From the Super Mario Wiki
The Nintendo Coin is a silver coin depicting the Nintendo 64 logo, and it appears in Donkey Kong 64. It and the Rareware Coin are required to retrieve the final Boss Key located behind a door in Hideout Helm.
To obtain the Nintendo Coin, Donkey Kong must play the original Donkey Kong arcade game in Frantic Factory. After completing it once, Donkey Kong earns a Golden Banana. Squawks appears in the room and notifies that he can win a special prize by playing it again. After paying a one-time fee of two Banana Coins, Donkey Kong can play the game again, at a higher difficulty level. Instead of rescuing Pauline, the player must rescue the Nintendo Coin from Donkey Kong.