One of the harshest decrees the Greeks made against the Jews was that they were forbidden from studying their holy Torah. One way they managed to study in private was to have children play with spinning tops (dreidels) outside the places of learning and this would be a distraction for the Greek army enforcing their laws.
A deeper source says that this dreidel also symbolizes the nature of the miracle. On Purim we play with the grogger. Since the miracle of Purim was a hidden miracle the grogger symbolizes this by having the hand below turning the grogger from underneath, in a hidden way, as G-d turned the events in a hidden fashion.
However on Chanukah we are celebrating the open miracle of the story, so the dreidel represents that by having the hand on top turning it around, it symbolizes G-d turning the events of the story in an open and revealed fashion.