In The Legend of Zelda: The Minish Cap, the hat that gives Link advice is named Ezlo.
Ezlo is a magical, talking cap who accompanies Link throughout the adventure. He can speak, offer hints, comment on locations and characters, and help Link use special portals that shrink him to the size of the Minish. This shrinking ability is central to the game's puzzles, because ordinary objects become enormous obstacles or useful pathways when Link changes size.
Ezlo is not merely an enchanted accessory. He was once a Minish sage who created the Mage's Cap, a wish-granting hat. His apprentice Vaati stole that artifact and transformed Ezlo into his current cap-like form as punishment for opposing him. Ezlo's story therefore connects directly to Vaati's rise as the game's villain.
Players sometimes confuse Ezlo with Navi, Fi, or Midna because all four serve as memorable Zelda companions who advise Link. However, Navi is from Ocarina of Time, Fi is from Skyward Sword, and Midna is from Twilight Princess. Ezlo is the talking cap and companion specific to The Minish Cap.