A "byword" may be defined as "a widely-believed saying that represents a known real-life situation." It is used to translate a number of Hebrew words, all of which mean the same. In the case of God's people (ancient or modern), just as the LORD blesses those who obey Him, so too does He make a "byword," a public example for all the world to see, out of those who fall away from His Law and become a lawless law unto themselves.
"28:15 But it shall come to pass, if thou wilt not hearken unto the voice of the LORD thy God, to observe to do all his commandments and his statutes which I command thee this day; that all these curses shall come upon thee, and overtake thee ... 28:37 And thou shalt become an astonishment, a proverb, and a byword, among all nations whither the LORD shall lead thee." (Deuteronomy 28:15,37 KJV)
"An astonishment, a proverb, and a byword"
The quote above refers to the time of Moses and the Exodus (see A Journey Without A Destination to understand why it took 40 years for the Israelites to walk from Egypt to the adjacent land of Israel), but the principle never changed. The LORD (i.e. Christ; see 'Before Abraham Was, I AM') displays His people in whatever way that they behave toward Him, blessings for good, destruction for evil.
In the time of Solomon:
"9:4 And if thou wilt walk before me, as David thy father walked [see also David, Future King Of Israel], in integrity of heart, and in uprightness, to do according to all that I have commanded thee, and wilt keep my statutes and my judgments: 9:5 Then I will establish the throne of thy kingdom upon Israel for ever, as I promised to David thy father, saying, There shall not fail thee a man upon the throne of Israel.
The same "good fruit" or "bad fruit" lesson is found also in Jeremiah (as it is in practically every other Book of the Scriptures). In this example, the original Hebrew word translated as "byword" in the verses quoted above is translated as "taunt" here i.e. "a reproach and a proverb, a taunt and a curse" (Jeremiah 24:9 KJV)
"24:1 The LORD showed me, and, behold, two baskets of figs were set before the temple of the LORD, after that Nebuchadrezzar king of Babylon had carried away captive Jeconiah the son of Jehoiakim king of Judah, and the princes of Judah, with the carpenters and smiths, from Jerusalem, and had brought them to Babylon [see Why Babylon?].24:2 One basket had very good figs, even like the figs that are first ripe: and the other basket had very naughty figs, which could not be eaten, they were so bad.
Fact Finder: What happens when Christian-professing people claim that God's grace is a license to not obey His Law?