This element was named after Tantalos, a character out of Greek mythology. Tantalos was a powerful king, who was punished by the gods for commiting several crimes. He was forced to stand in a lake up to his chin. But whenever he tried to drink from the water, it disappeared. Delicious fruits were hanging above him, but whenever he tried to reach for them to eat, they rose beyond his reach. This became the source of the English word tantalise.
The chemist Anders Gustav Ekeberg who discovered the element, tried to dissolve tantalum oxide in acid, which did not succeed no matter how much or how strong the acid. This reminded him of the situation of Tantalos, who couldn't drink the water he was standing in. Therefore the name.