Besides the illustration, referenced by Cosmo above, on the NG Discussion Forum from John White's "The American Railroad Passenger Car" that shows similar rings on a standard gauge coach of more-or-less similar vintage to the two original B&SR coaches, an internet image search (depending on your search engine and search terms) will reveal other examples of such rings, sometimes incorporated as part of luggage rack end castings. I've attached an example, from what appears to be a museum setting, in which hats are shown hanging from some of the rings. Even so, I'd argue that this may be a misinterpretation of the actual purpose, as rings mounted directly on the walls in the B&SR example would seem ill-suited to such use. In fact, in the illustration from White's book, the rings appear to part of the bottom of the rack castings, clearly impossible to hang a hat from.
Cosmo may have the right idea regarding umbrellas - even though someone on the NG Forum rejected that interpretation because hanging umbrellas would swing and mar the woodwork! Clearly that criticism misses the point that the umbrellas could have been slid horizontally between two rings. In addition to Cosmo's umbrella idea, one may note from many old photos that walking sticks and canes were once in common use. (When I was a kid, walking sticks were many decades out of fashion but quite a few elderly people walked with canes; nowadays I suppose they get hip replacements instead...)
Especially given the examples of such rings being incorporated into luggage racks, I suggest that the path to an answer isn't imaginative speculation about gloves or even miniature spittoons (!), nor even "scientific method" processes of elimination. Rather, we should realize that railroad car builders did not typically manufacture the fittings (such as lamps, luggage racks, and seating) themselves, but purchased them from suppliers of such items, such as Hale & Kilburn or Heywood-Wakefield for seats, and Adams & Westlake (later Adlake) for racks, lamps, locksets, and other items. While I don't have access to an Adlake car trimmings catalogue for the era in question (or for the somewhat later era suggested by the attached photo), such resources exist. I would assume that catalog images of luggage racks with the ring and others without it would be accompanied by text descriptions of the functional differences and relative advantages. Such a catalog might also show individual rings for use without luggage racks, as in the B&SR example.
