(Linking back to
the previous thread for reference once they get separated.)
If we assume that everything Tolkien wrote is 100% correct unless explicitly replaced (which... we shouldn't, but I always do

), then it is clear that Gandalf was looking for information on the Ring. LotR is nearly explicit that he did not believe Bilbo's Ring was the One until immediately before that final visit in 3017, so why was he studying the Ring?
I wonder if the reason is almost
too obvious: because Sauron had openly returned and the Nazgul were at large again. The idea of Sauron getting the One Ring back must have haunted the nightmares of the Wise, and Elrond had already (per the Silmarillion) told Gandalf of his foresight that the Ring would return. That meant Saruman's airy reassurances that it was lost in the Sea could be ignored, and so... where
was it?! The only place that might have records on the subject was Gondor.
The biggest concern was probably that the Ring had been
found. If I were Gandalf, I would want to check Gondor's archives to see if there were any rumours of possible Ringlords in the Third Age, particularly the first millennium or so before Sauron re-emerged and the Istari arrived. An Orcish or Mannish chieftain could have found and used it, established a brief dominion and then lost it through the undying malevolence of Sauron. (The thought that someone would claim the Ring and use it only to catch small goblins to eat probably seemed impossible.)
I also wonder if he was hoping to find some way to
destroy the Ring, if and when it emerged. Elrond said it was impossible - but the Numenoreans, by many accounts, surpassed even the Noldor of old in skill, save it be Feanor himself. They successfully invaded Valinor, and even after Numenor fell they went about building walls and towers stronger than Gondolin ever was. If anyone had figured out how to destroy the One Ring, it would be the Numenoreans. A fool's hope, perhaps, but one that was worth looking into.
hS