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, and Squawks appears in the room. When talked to, Squawks informs Donkey Kong that he can win a special prize by playing it again for two Banana Bunch Coins. The second playthrough is more difficult, requiring the second level to be completed with only one life; the reward is the Nintendo Coin, which replaces Pauline at the top of the stage.