Q. Why do we kindle Chanukah lights in the synagogue as well as at home?
It is more likely that the practice was instituted for the sake of pirsumei nissa, publicising the Chanukah miracle.
Some sources say that kindling lights in the synagogue is for historical reasons; since Chanukah marks the restoration of the lights in the Temple it is appropriate to kindle the lights in the synagogue.
One might also quote the mystical notion that the Divine light needs to be brought out to the world, and by means of the menorah in the synagogue, Jews rededicate themselves to the search for the fullness of that light.