Longer version:

Palantíri (singular, palantír) are the "seeing stones" from Lord of the Rings. They're kind of like mythical crystal balls, but with greater powersets. They were designed by the high elves of Valinor (read: uber-magical), and can be used for communication. For more powerful entities, like wizards, they can be used to see anywhere in the world. Palantíri are rare, powerful and incredibly useful in the right hands.

Saruman uses his palantír to great effect to keep track of the various going-ons of Middle-Earth, and from his tower in Isengard is able to make tactical decisions based on information gathered by the palantír.

Given the stones' incredible power for accumulating information and forming strategic positions based on that information, Palantir Technologies chose the name as a semi-obscure reference to a mythical object of great power. This allies itself well with what Palantir represents - powerful data analysis and mining tools (among other things) with a beautiful design and focused precision.