The New York Criminal Law and Procedure blog tries to answer what is the “interest of justice”? Courts in New York are authorized by the Criminal Procedure Law to disregard certain rules only in the interest of justice, but it’s undefined and (I think) rather vague.
Larry Cunningham took a stab at it in an article about the appeals process. It’s a good read, and you really don’t want to be outside on the East Coast right now, anyway.