The folk in Lake-town told me as Dale might be better for knowing things from the past, but after couple months of visiting it -- it took that long! -- I learned that they rebuilt the city even more recent than Lake-town, and it were destroyed even longer. They do got libraries, and they been building them back up as they can, and their scholars are more learned since the city's not only about trade -- but even so, it's a new city built on top of the old one, and no one alive remembers very far back, so it's no surprise they couldn't really help.
I don't know if it's a thing about Dale, or just the librarian what I got talking to, but the price to get answers weren't coin nor any other trade, but something most odd. Instead, what he wanted was for me to tell him everything I know, and to tell it in proper Westron. And he were mortally offended by the way I talk. He weren't even too pleased by my Rohirric, but at least in Rohirric I mostly talk proper. But I guess the way I learned Westron is all full up with ignorance of proper rules. So he gave me terms what, when I heard them, I thunk as he were joking. But no, he meant it. He'd teach me some proper Westron, and then I'd use it to teach him about the history I knew. Then he'd teach me a bit of letters, both reading them and writing, and then he'd teach me some history as he thought might help with my search, or help me use my letters to find books and scrolls what might have answers. Then a couple days later I'd come back and we'd do more. And he wouldn't answer all my questions until I could speak Westron as proper as he wished.
So now, I can speak properly when I wish to. It is hard, and takes a lot of effort. When I don't think about it, I lapse back into my normal way of speaking. But it's useful for making a good first impression, so it may be helpful when I first attempt to speak to the Dwarves of Erebor, which is what I plan to do next.
From the librarian and his books, I learned a lot about the history of the Wilderlands, which is what they call all the land from the Misty Mountains to here. I learned about the days afore Eorl, with the Northmen and the times they went to war with the Easterlings, and how that led to Eorl's ride to the aid of Gondor against the Balchoth. I learned about the chieftains afore Eorl, and how Framsburg were named for Fram, what slew a dragon -- I guess dragons are a real thing after all, or were -- called Scatha. I learned about how the Northmen became, mostly, the Woodmen, excepting them as followed Eorl, and some what joined with Beorn after the Battle of Five Armies to patrol the lands. I even learned a bit about the lands west of the Misty Mountains, like the kingdom of Arnor what used to be sister to Gondor afore it broke into pieces -- he had me learn that on account the lantern might come from Númenor and what got brung there got put some in Gondor and some in Arnor.
But on the lantern, I don't really know anything I didn't learn from the Woodmen, I just know it a bit more sure. The folk of Eorl's what didn't join him in the Mark almost certainly went to dwell with the Woodmen and got folded into them. Either the lantern went with one of them and got lost along the way, or there's a small number of them what took the lantern and went a different direction. But there's no mention of a lantern in any record the librarian could find, nor of any other movement of the Éothéod. Nothing as gives me a direction to look.
I need to find a way to speak with the Dwarves of the Lonely Mountain next. They may not know nothing about the Éothéod, but they might know about the lantern. Either how it were made, since they know all manner of things about crafting, or what might've come of it. It's even possible the lantern came to them in trade. Thing is you can't just walk up and ask to be let in. I got to find a way, back in Lake-town, as I can talk to the Dwarves.

