The aircraft lined up for takeoff in conditions of reduced visibility. The crew believed that the lights they could see ahead were runway centreline lights when they were actually runway edge lights. The aircraft began its takeoff run but ran off the paved surface and onto grass. The commander closed the thrust levers to reject the takeoff. Information available to the pilots allowed them to develop an incorrect mental model of their route from the holding point to the runway. Environmental cues indicating that the aircraft was in the wrong position for takeoff were not strong enough to alert the pilots to the fact that they had lost situational awareness. One Safety Recommendation has been made.