How come Holden says "goodby" instead of "goodbye" in Catcher in the Rye?

This is an interesting question for an interesting novel. Salinger's The Catcher in the Rye takes on a three-day journey that ulitmately leads the protagonist Holden Caulfield and his readers nowhere.

The story is set in the late 1940s—1948 or '49, depending on when Holden's birthday fell. Author J.D. Saligner worked to make a novel have the authetic feel of the time period. Interestingly, the spelling of "goodby", as used in the book, was a common, though not the most popular, way of spelling "goodbye" at the time. Spelling it as such lends that that authenticity. 

Looking deeper into the implications of the word, "goodbye" is a mutation of "God be with ye." Certainly, Holden is grappling with the role God plays in his life and offering this phrase may cause him to feel more like the phoney adults he deplores. The alternate spelling of "goodby" is perhaps a subconscious way of sticking it to the man, so to speak. Holden struggles with the nature of goodbyes and by subverting the word, he can minimize its meaning in religious sense and in a rebellious teenager sense.

However, whether or not Salinger meant it this way, we will never know.