"Interest of justice" is used in a couple of places in the CPL. One is for a dismissal of an accusatory instrument at the trial level. The test is well established and there is abundant case law explaining when a trial court should and should not grant a motion to dismiss in the interest of justice.
There is another mention of the term in the CPL: Intermediate appellate courts (Appellate Division, Appellate Term, County Court)—but not the Court of Appeals except in death penalty cases—can reverse or modify a judgment in the interest of justice. This allows the courts to reverse or modify for unpreserved questions and to reduce a sentence that is lawful but excessive. Unfortunately, there is little guidance for courts to determine whether to exercise this discretion.
In a recent article, I explored the interplay of preservation and "interest of justice" jurisdiction. I compare the federal and state standards and offer a test for courts to apply. The article can be downloaded from my SSRN page. (LC)