Corrected entry: When the English girl phones the black guy her name comes up on the mobile phone, but the phone should have only read a number, as the phones are set to call out only to the 4 digit numbers on the white masks.
Corrected entry: At the beginning of the movie when Mike shows up at Chelsea's apartment wearing a Chatterer mask, Chelsea says that she didn't open "The Lamont Configuration" referring to the puzzle box that call forth the Cenobites. The correct name of the box is "The Lament Configuration", as it has been called in the previous movies and in all Hellraiser literature.
Correction: Character mistakenly mispronouncing the name is not a movie mistake.
Correction: The phones can easily be programmed to display a name for call display purposes. Lance Henriksen's character knew who was coming to the house, and would have had plenty of time to program those particular phones. Besides which, since much of the time spent at the house had the main characters hallucinating after being drugged, they could just as easily been hallucinating their own names being displayed.