White Mouse was sent by the Princess to assists Prince. The mouse found Prince just as he was in trapped situation, and the mouse helped him by stepping on a pressure plate.
The same mouse was still with Princess, after Jaffar had taken control of Persia. The mouse sadly sat with the Princess trying to wake her up from the magic sleep, until the Prince defeated Jaffar, and Princess woke up.
White Mouse was intended to have a bigger role in Prince of Persia 2: The Shadow and the Flame. White Mouse sitting on the ship would have been a hint for the player to jump on that moving ship. On the Ship there were storyboards planned of Prince interacting with the White Mouse. The mouse was also how the Princess would recognize the real Prince when the mouse climbs from his hand to his shoulder.
White Mouse in Prince of Persia SNES
White Mouse with Princess in Prince of Persia SNES