A New York bar's website is one of a few sites which provide the following (seemingly) credible explanation for this practice:
No one is exactly sure of the reasons why larger format bottles were given biblical names. But, according to the Champagne expert Francois Bonal, winemakers in Bordeaux had been using the name Jeroboam for the four-bottle size since 1725. (It's presumed they selected Jeroboam, the biblical founder of Israel who ruled from 931-920 BC because he is referred to as "a man of great worth," as were the larger sized bottles). Bonal also explains that a Champagne Medieval poet, Eugene Destuche, mentioned several of these names in his poetry. The region of Champagne adopted the Jeroboam size and followed suit with larger format bottles developed in the 1940's, continuing the practice of selecting biblical kings and patriarchs.
François Bonal does appear to be an expert on Champagne and has written a number of books on its history. The stated explanation was very likely taken from his book, Le livre d'or du champagne (1984).
A similar theory is proposed by a columnist in an Alabama newspaper:
The earliest recorded use of a biblical name for a large-format bottle comes from Bordeaux, where in the early 18th century winemakers called a large bottle holding four standard size bottles a jeroboam, for the first king of Israel. Jeroboam is described in the Bible as a man of great worth. Hence a large bottle of Bordeaux would be of great worth, but this premise is scholarly speculation.
As noted above, the "man of great worth" reference is from the Old Testament's Kings (which is in itself suggestive):
Now Jeroboam was a man of standing, and when Solomon saw how well the young man did his work, he put him in charge of the whole labor force of the tribes of Joseph.
The above is just one of seemingly myriad translations. Solomon's son and successor was Rehoboam. So I guess there's an element of proximity at play as well.
(There might be something in the fact that these names are taken from the book, Kings, as well as the fact that all of them were, along with a number of others in the list, all kings. Even Balthazar and Melchior of the Magi are apparently considered to have been kings.)