The Thirsty Lady is a character found in the Town of Nabooru in The Adventure of Link. Like the Mirror Lady of Saria, she wants Link to perform a simple task and bring her some Water. He can find some by searching a nearby fountain in the same section of town. Carrying the water back a short distance to the girl will gain Link access to the Wise Man, who teaches him the Fire Spell.
Not every task takes Link to the depths of Death Mountain.