So, thanks to Steve Morrison's comment on my earlier post* (thanks Steve), I've now learned about another Ring of Invisibility, this one occurring in E. Nesbit's THE ENCHANTED CASTLE . Turns out (thanks again, Steve) the first to point this out seems to have been a Dean Hazelfan back in 2003:
More recently (Sept. 13th), Wayne & Christina have posted a more detailed account of the book's magical ring and the specific ways it might have influenced Tolkien (thanks Wayne):
Now that I've read the book for myself, I agree that it might well have been another element in the pot that contributed to Bilbo's ring. The most distinctive shared feature, as Wayne and Christina point out, is that Nesbit's ring makes the wearer invisible but not her shadow, which creates awkward situations. There are some minor points that may also be worth mentioning, such as one character's constant concern for missed mealtimes, the difficulty of getting meals when one is invisible; the breezy narrator is also at times rather like that in The Hobbit.
One interesting point is that Nesbit's is not really a ring of invisibility at all; rather, it's a ring that causes whatever its wearer says to come about. Thus at first it makes various of the characters invisible; then it makes one an old man, then another twelve-feet tall, then another a statue; eventually it brings two lovers together and creates a ghost. While it does at first seem tricksy, eventually the children learn how to use it properly; it's not in any way sentient, or evil, though its effects usually cause terror in all viewers.
It's not as good a book as FIVE CHILDREN AND IT, which we know was a direct influence on JRRT (the 'It' reappears in ROVERANDOM), nor an important influence on THE HOBBIT, such as Kenneth Grahame's THE WIND IN THE WILLOWS and Loftings DOCTOR DOLITTLE books. But it's definitely worth considering as another element in the mix.