Some are more or less obvious, and some have voice lines when some champions purchase them.
A lot of items reference a place or a person, but don't really explain about those. Things like 
.
And then there are Doran's items... but Doran doesn't exist anymore in the new lore, I think?
Many items are named after designers, some are named after fans, and some are references to other games. Lore about them is sacrce but some items do have proper history behind them while others have elements to their design or comments made by champions that inform their place in the lore:
the most famous example, the blade of the man who caused the Ruination. The blade itself doesn't have much history, but the man who wielded it does.
Worn by Gangplank, and given the name and quote, it stands to reason that Gangplank killed its previous owner and took the shoulder pad for himself. It's a very traditional Bilgewater armour piece given that it's styled after the maw of a sea creature.
Worn by the Seeker, that being Lissandra, when that name meant something. As the Seeker she was the voice of the Frozen Watchers, relaying their orders to the Iceborn. She shed the armour that the armguard belonged too long ago and it was later reclaimed by Sejuani, who wields it today.
This one's easy, as it's a Kalista only item; this was the weapon that ended her life and the one used to enact pacts with her.
Bears the crest of Shurima very evidently, placing it as a relic of ancient Shurima. Interestingly, Azir's soldiers have a capes that are remarkable similar to the Sunfire Cape, so perhaps it was something worn by the emperor's elite guard before the Empire's fall.
all True Ice artifacts or relics of the ancient Freljord. FFQ is likely tied to Avarosa as she's the only character we know of who would've bore a title such as Frost Queen. Guardian's Horn, Hammer and Orb are presumably owned by Greyor since he wears the horn in the lore. I like to make a case for 
also being True Ice but that's just me.
Given the existence of sea creatures such as hydras around the Blue Flame/Serpent Isles, it's likely that these two weapons originate from there.
Both bear the crest of Demacia, placing them firmly in the armouries of that nation. This also puts Randuin in to context as a Demacian.
All artifacts of Mount Targon, though Eye of the Equinox is more closely tied to Targon due to its form and design.
Given the spectral copy blade's form, it's safe to say this is a relic of the Shadow Isles. The same can be said about 
Ekko mentions in his item interactions that the Revolver and Gunblade are Piltovan weapons, not Zaunite, which implies that GLP-800 and the Protobelt are Piltovan as well.
All seem Demacian too me, with Righteous Glory and Crystalline Bracer being similar to Garen's armour, Redemption looking like the crest of Demacia in some ways, and knights being something almost exected in Demacia in a major way comparatively to other parts of Runeterra.
Right I may come back to this later with more item info I can conjour up. Keep in mind a lot of this is conjecture. I encourage you to extrapolate based on the bits and pieces floating around in game.
I would guess Doran still exists in the lore. They retconned him, and his lore isint reliant on the Institute of War, so his story is probably intact.
