You must understand George Burns (born Nathan Birnbaum; 1896-1996) in context (and the context of his beloved wife, Gracie Allen (1895-1964)). George Burns lived forever, and more importantly, he remained on the stage forever. He was the Betty White of his time. The older he became, the more cherished he became as a national institution.

George and Gracie were a vaudevillian comedy duo who began performing together in 1923. The brilliant Gracie played a (sadly) ditsy wife, always having the best lines to George’s deadpan delivery. As their signature closing, George would say, “Say good night, Gracie.” And Gracie would answer, “good night”; in the context of the act, Gracie was too simple to know when to close the performance, so in her silly fashion (as part of her ditsy character), she was following the direction of her husband. For sexist twentieth-century audiences accustomed to the cliche of the dumb wife, this exchange of lines made sense and was considered funny in itself.

George always credited his success to Gracie’s comic brilliance. George was the straight man with a slow, dead pan delivery. With his signature cigar, he delivered witty, but droll, one liners in the style of Mark Twain (who was barely cold in the ground when George began appearing on stage). Gracie provided the high notes, capturing the audience’s attention and most of the laughs.

Because Gracie died at a relatively (compared to George) young age, and because George persisted as a solo performer, there was plenty of time for him to remain in the public eye and for this farewell dialogue to mutate in American tradition; later folklore maintained that her response was often, “Good night, Gracie” – a further expression of her ditsy character. Apparently, this did not really occur, however. She consistently responded with a simple, “Good night.”

George continued to perform with his signature deadpan sense of humor, appearing on his hundredth birthday (there was a similar buildup to his centennial that was expressed and hoped for in the case of Betty White who died a few months ago, just short of her 100th). That meant that George had over thirty years of American performance, particularly on American television, after Gracie died, and he always closed his act by saying, “Say good night, Gracie” or merely "Goodnight, Gracie," looking up towards heaven.

The phrase was taken as a loving tribute to an old comedy act split apart by the death of a beloved spouse. It was poignant and comic, and everyone understood its context because George persisted as a national icon. By perpetuating the phrase, George kept the national memory of Gracie Allen alive, long after her final performance in the early 1960s.

The phrase has little meaning today, and it did not exist before George and Gracie invented the dialogue in the 1920s. Because it persisted in American comedy for seven decades, as long as George lived, the phrase remained part of the national vocabulary.

edited to add an explanation as to why the line was regarded as funny.

edit #2: To understand why the Burns-Allen act was considered funny, one can look at the analogy in the TV program, "I Love Lucy" (1951-1957). This program exploited the same theme of a ditsy wife and a steady straight man who provided the frame for the silly antics of the wife to strike similar, sexist comic chords. The Burns-Allen act was, of course, exploiting these themes decades before the Richardos in the 1950s.

edit #3: I found online testamony from people who swear they heard Gracie occasionally say, "Good night, Gracie." The problem here is that without hard evidence that this occurred (if it did occur, I'm sure it is out there somewhere), we must also concede that folklore can affect memory, and we can believe we heard something in the past based on what tradition tells us occurred. In addition, in "Rowan and Martin's Laugh-In" (an NBC television show, 1968-1973), Dan Rowan (the straight man) would often close the show by saying to his goffy partner, Dick Martin, "Say good night, Dick," to which Martin would reply, "Good night, Dick." This could have reinforced the idea that a similar response from Gracie was usual.