In the scene where the two janitors first enter Michael's room when he is an adult, when the shot shows Michael and all of his masks, one of them in the middle left resembles the mask from the original 1978 version.

Aside from the more obvious relations between the original and the remake (i.e. "Don't Fear the Reaper", etc), one interesting one is how Michael acquires his trademark jumpsuit. In the original, it's implied he kills a trucker and steals his (we see his dead body in a field left with Michael's robes). Sure enough, in the remake Michael also gets the jumpsuit from a trucker - of course this time the death is plainly visible.

Danielle Harris, who plays Annie Brackett in the movie also played Jamie Lloyd in Halloween 4: The Return of Michael Myers & Halloween 5 The Revenge of Michael Myers.