The LEGO Crossover series (LEGO Star Wars series, LEGO Batman, etc.). They seem to work because, well, they aren't rushed to be released at the same time the movie comes out. Though the first LEGO Star Wars game was intended to come out at the same time as Revenge of the Sith, it also covered The Fantom Menace, and Attack of the Clones. The second one came out decades after the movies that it covers, the third one is just the first two combined into one game, and the fourth one was based on expanded universe material. LEGO Indiana Jones was a game based on the first three movies set to release a couple of months after the fourth one, and the second LEGO Indiana Jones was just the previous two plus the fourth movie. LEGO Batman has an original storyline, though launched within what, a year of The Dark Night (Ironically, LEGO Batman is closer to, say, The Batman (More serious than the Adam West series, but a bit more lighthearted than the 90's animated series) than it is The Dark Knight)? LEGO Harry Potter was based on the first four movies/books and was launched about six months prior to the seventh movie.
Also, there's the Harry Potter game on the GameBoy Color, which is a turn-based RPG that lets you explore a bit, and includes stuff that was left out in the transition from book to movie, essentially making it "Harry Potter: The Game of the Book", as opposed to "Harry Potter: The Game of the Movie of the Book" like the other Harry Potter games.