Apparently "of a sudden" comes out of Shakespeare. The phrase "all of a sudden" is a derivative of that. "Of a sudden" is a preposition, "a sudden" referring to a nondefinite noun. "The sudden" is a definite noun, thus changing the implications of the phrase. If one means to say "All of the sudden," it is only grammatically correct to say "All the sudden" if "sudden" is somehow a plural noun. Which it is not.

"Sudden" refers to a sense of time. The time is "a sudden." Did the action take place over the course of "all of a day" or "all of a sudden?" Why, it happened all of a sudden!

It is still an idiom, but "all of a sudden" is proper.