The word is in the sense of news, tiding, gossip, rumor.
Partridge’s Dictionary of Catch Phrases has it, giving a citation from John Kendrick Bangs’s 1913 A Line of Cheer For Each Day o’ the Year, in the entry for 23 April:
What’s the good word?
Now that’s a phrase I truly love to hear,
And when ‘tis heard,
I always smile and promptly answer “CHEER!”
It holds more warmth and genial glow
Than any other word I know.
From this we can deduce that the phrase was fairly common in 1913 and people were asking the same questions about it.
I also found this in an article titled “Broad Street Gossip” in The Wall Street Journal, 24 Nov 1914, 2, that shows the “news” meaning of word:
A banker who has taken a prominent part in solving the difficulties brought about by the European war was met by another of about equal prominence late on Friday afternoon and was asked: “What is the good word?” His reply was that the outlook for the next day’s weather was excellent, and it brought the immediate rejoinder: “That is fine; I am going too.” Each had in mind the Yale-Harvard football game, which both regarded as the most important thing the in prospect.