I'm gonna cheat a little with Demon's Souls, but the rest will be more traditional answers. I hope it's ok!
Demon's Souls
Silver Catalyst
I've never been into magic in games, always went with melee whenever given the choice, and Demon's Souls was no different. Did my first playthrough with a Claymore, and kept using melee without really thinking too much about it. Flash forward to 2018 (I think? Maybe 17), and for whatever reason I can't explain I decided to give magic a try, starting with Demon's Souls, and it blew me away. Not even the gameplay itself, though it was incredibly fun, but how brilliantly this specific game ties sorceries and miracles with the story, and how much deeper and more interesting the lore gets when you start interacting with the NPCs that never even talked to me before because I didn't have the stats and didn't really care.
Starting that mage playthrough not only opened up a world of possibilities in gameplay, making me play every game in the series as a sorcerer and later as a cleric (though Demon's is still the most fun for sorceries, imo), but made me completely fall in love with Demon's Souls in a way I never did before. It was always my least favorite in the series, or at most beating Dark Souls II, couldn't really decide, but after that playthrough, it quickly climbed my list and now sits comfortably as my third favorite, only behind Bloodborne and Dark Souls 1. It's a brilliant game, and manipulating the Soul Arts ties your actions to the story in a way that truly elevates the experience.
It's interesting how it was actually the first Souls game I played, back when the US version came out, but it only really clicked with me many years later.
With that said, it's only logical that my second choice for this game would be the
Talisman of Beasts
An old wooden amulet resembling the Old One.
It can utilize both miracles and spells.
The symbol of God was nothing more than the Image of the Old One.
Such a simple, yet powerful item description. A haunting twist that brilliantly ties the conflict between clerics and sorcerers with the game's central theme and plot. What a masterpiece this game is. 10/10
Dark Souls
Reinforced Club
I know, I know, the most boring ass answer in the world, but this was the weapon I used as I fell in love with this series. As I mentioned before, Demon's was actually the first one I played, but it was with Dark Souls 1, in 2013, that the series really clicked with me, and it became my favorite franchise in gaming. This weapon taught me about scaling, as it slowly got better than that shiny Drake Sword and its higher stats that kept tempting me to go back to it, it taught me about exploring the moveset some more, with its forced jumping attack on R2, taught about how Souls games are balanced and mostly fair, with this shitty-looking weapon from the first real area of the game carrying me through the entire thing and the DLC.
It's mostly an emotional response about the fond memories I have of my first playthrough, but I do find myself going back to it quite often, and it's always a blast. I'd say that it's my favorite weapon in the entire game.
Zweihander
Ultra Greatswords are so cool. This one you can get very early in the game, has low stat requirements for two handing, is really strong but not quite Black Knight tier, it's the complete package. Each two handed hit just feels so damn good. The animation gives perfect feedback for this weapon. I LOVE doing Lightning Zweihander runs. Not any lore reason or emotional connection to a fond memory, this weapon is just simply awesome to use.
Dark Souls II
Greatsword
This picture is actually of my own character. Just look at this shit, it's too fucking sick! A lot of people tend to go with the Crypt Greatsword for their Berserk weeb needs, but I just like the way this one looks a lot more. Very fun to use, too, it's one of the very few weapons that I find the Dark Souls II version a lot more interesting than the original. Loooooove this shit.
It also helps that it's not that great, unlike the Zweihander in the original Dark Souls, since II's bosses are a bit on the easy side. Fighting Fume Knight with the Greatsword was amazing.
Dark Souls II is still probably the best in the series for Fashion Souls, too, and I really enjoy that aspect.
Short Bow
Another boring ass answer, but this one is kind of similar to the Reinforced Club story. When Dark Souls II came out and I got the Short Bow very early on, I realized that I never even once used a bow in the first game, and decided to change that. I was forcing myself to use it, even when it didn't feel particularly effective. Just kept it with me, since I had three weapon slots in this game anyway, and used it to snipe some enemies, lure them one by one, this kind of thing. Then, when reading discussion online and seeing the hatred Shrine of Amana was getting, I realized just how incredibly useful this thing proved itself to be. I never really used it much for bosses or anything like that, but just the fact that it made a huge impact in the way I perceived a lot of the game's areas and encounters in a positive way is remarkable.
Unlike the Reinforced Club, though, I didn't stick with this one for the whole playthrough, I used a bunch of different bows, but since this was the one that got me started, I'll mention it.
It's kind of hard to go back to pacient playstyles after a first playthrough, though, so these days I also hate Amana.
Bloodborne
Burial Blade
My favorite Souls game, and my favorite story in a Souls game, and yet, I don't give a fuck about the lore here, it's just a cool ass scythe. I love edgy scythes in anime, and I couldn't WAIT to finally find this weapon I've been missing that I saw in gifs and screenshots. I saved my single Blood Rock for it, I would go search how to get it after finishing the game, because I didn't want any spoilers, so when I finished the game and saw it, I immediately upgraded it to +10 (finished the game with Burial Blade and Threaded Cane) and went stragiht for a NG+. No pause, no time to discuss the game or whatever, just straight through NG+, got another ending, and then started the late-game Chalice Dungeons, all with this weapon.
Previous scythes in this series have always been kind of lame to me, but they really hit the mark in how edgy it should be. It's the best. The charged R2 gives me life, as seen in gifs posted here.
Holy Moonlight Sword
Let's be honest here, everyone knew it was coming. Before they even announced any DLC for this game, everyone knew two things about it: We'd fight Ludwig, and he'd use the Moonlight Greatsword. Still, I wasn't ready for how incredible both the fight and the sword would be. I wanted it for tradition more than anything, but it ended up being one of my favorite weapons in the series, it's just SO much fun.
The arcane damage is kind of irrelevant, but still made me glad that I upgraded that shit on my first character just to see what the Hunter Tools did. It's so beautiful, and Ludwig is such a tragic figure with an interesting lore that isn't explored that much in the game, so having that sword with me just kept me thinking about it all the time. My true mentor, my guiding moonlight.
This is peak Moonlight Greatsword, it can't get any better, that's why I'm not even sad there won't be one in Sekiro.
Dark Souls III
Onikiri and Ubadachi
Really enjoyed the twin weapons in Dark Souls III. Started as a Mercenary on my first playthrough, and I just love them. Some are just horrible, like the Crow Quills, but I still love them even then. And the Sellsword Twinblades are insanely OP, they cut through bosses HP like it's nothing. Before the bleed nerf, that shit was CRAZY. Went with the katanas mostly because of the moveset, I love that jumping attack. Though I was also considering the Irithyll Straight Sword for this spot. It's a very simple straight sword, but oddly enough it was the weapon I used the most the first time through, and I used MANY weapons on my first playthrough of Dark Souls III.
Firelink Greatsword
This is a weird choice to me, because, to be honest, I don't really like this weapon or the associated boss all that much. If this was a Dark Souls 1 weapon, I'd never use it. But I guess just because it's the final Souls game it feels right to use it. There are many weapons I think look cooler, or I have more fun playing, but I often find myself just wearing the Firelink Set and using this one. I welcome Déraciné, Sekiro and every other weird game they decide to make, I think it was about time, but I'd be lying if I said I won't miss Souls games. I hope they come back, eventually.