Magic items are capable of a great many things. They can give players and villains alike a needed boost in strength. They can be the source of entire adventures, or trinkets that the players don't think twice about. However, they tend to get stories associated with them over the course of adventures. Even otherwise seemingly useless items may gain life for their utility, or be remembered fondly for the one time they proved to be useful. At this point they become more than just a magic item, they become a story. And it is this topic I hope to explore today.
More Than A +1 Longsword
An easy first step is to avoid giving +1 weapons. Make the item something special by adding a couple of details. Even something small like a few notched on the cross guard can lead to an interesting story when someone they run into finds it familiar. Now, I'd usually recommend something bigger but subtly can be surprisingly effective too.
In terms of function, I've found going big is most memorable. Something that changes the game. The real potential for story telling with these items isn't that it's just a sword like my old one but does more damage. It's that it opened up new options previously beyond the player's reach. The new options that open up allow for new stories to be told. However, you can do both aspects or one or the other. The great part about an item, even if it's just a +1 longsword in combat, is that it might open up role-play opportunities not otherwise possible. A +1 longsword in a world with almost no +1 longswords becomes something more. It will be coveted.
Let's Find Them!
There is always the classic story of trying to find an item. A whole campaign could revolve around finding an item. Whether that's because it's valuable to the players or the NPC that hired them is irrelevant: the item has value and they'll look for it. It takes care though, and players who are willing to go after such a goal.
However, I think that you can't rely on the item itself in this case to drive the whole campaign and story. Instead it's a catalyst for the players and forces who want it to perform actions. As such, the location where the item is and the path to getting to it is the real core of the story. Where and how an item is found, even one that wasn't sought, tells quite a lot about that item and can hint to a larger story. This may sound obvious but I've seen all too often that an item becomes all that is important in a campaign instead of a setup.
An item needs to be made by someone for some reason, and if the players aren't going to pawn it off at the nearest stand they'll need a purpose for it too. The purpose may not be foreseen, since it's up to the players to find creative methods to use items. However, if the players realize they are about to face a dangerous situation it isn't uncommon for them to look for ways to change the odds more in their favour. And a common method is through magic items. Scrolls, magic rings, world shattering devices, a sun blade in a vampire centred campaign, take your pick. What's great about these situations is that since the players are actively seeking these items out, it means they already have a purpose. Of course, the trouble with this method is that if the players always break to look for magic items and resources, a good portion of the tension will fly out the window. They can't always run into a situation perfectly prepared.
When it's on the other side of the DM screen I find that the items should help further the goal of the NPC in some way. If the NPC or villain is also chasing after the same item, they need to have a reason. Money is an easy one, but something more personal can often lead to better results. Potions to prevent aging, or the severed limb of a now crippled death knight who wants to regain his power. Those sorts of things.
Magic items can become part of the identity of a character. I think we can all remember a character we had that had a certain item that was closely associated with the character in question. Even if it's a published item like a sun blade, the moment it was used cleverly to kill 3 vampire spawn goes down in campaign history and leaves an impact. What it requires to become more in this case is a special moment and use for it. After all, a sun blade wouldn't have been nearly as fondly remembered in a campaign with no vampires, though it would still prove to be more useful than a regular old short sword. This also means that if a player character dies, their old magic items have a history attached to them. I've seen this occur both in cases where players reacquired items their old characters used to have (the campaign was set a thousand years after their previous ones), or where a new player joining a group received the items from their recently dead friend. The responsibility to do justice to their fallen comrade that the player role-played was something hard to describe. The reactions the players had to regaining their old items that now had a long storied history in the world was also hard to describe. The item became a character in the campaign itself, and it was like seeing an old friend NPC come back.
On top of that, some magic items can help tell the story of a character. It's not that uncommon to see a player character who is a disgraced noble or hero. Besides their magic sword, and memories of grander days, they have virtually nothing. You can also have the item grow in power over the course of the campaign as the player becomes more powerful and gains more control over their magic item. In this case, their starting weapon is closely linked to their personal story and has a history all their own. It's also a history you can let the player come up with. Whether it's the magic sword of their fallen friend that they will take to finish their work or whatever else, such a history is a great opportunity for telling stories and role-playing.