2 - A wizard did it. On purpose, that it.
3 - The weapon comes from another realm/world/dimension/plane/reality, where things just work differently. A weapon (or other object) from here would probably behave "strangely" relative to the surrounding laws of science/magic over there, too.
4 - The weapon's magical properties are the unintentional result of some other phenomenon. It could be the byproduct of a spell or experiment, or it could have just been in the right place at the right time when some magical phenomenon ("natural" or otherwise) occurred. Maybe someone just left this weapon on a ley line and forgot about it for a hundred years.
5 - The weapon was forged from a rare material with magical properties. It could be mithril or Valyrian steel or adamantium or metal forged from a fallen star or whatever.
6 - The weapon has been used to kill so many supernatural beings (or such a powerful supernatural being) that some magic has rubbed off on it.*
7 - The weapon is only magical in a certain context. Silver weapons hurt werewolves. Oak stakes hurt vampires. A ghost might be banished by the same weapon that killed their corporeal form, due to their lingering fear of it. In short, the weapon has magical effects in certain circumstances due to the properties of the being under attack, or the "natural" relationship between the weapon and a supernatural force.
8 - See Result 6.*
9 - The weapon was forged with a rare technique which imbued it with supernatural properties. The creator of the weapon was not a wizard in the stereotypical sense; they were just such a master craftsman or such a knowledgeable expert that they knew how to make the weapon supernaturally extraordinary. This could involve a ritual, a blessing, another magic item, advanced technology, magic runes, a crafting technique that is just so darn good it causes "magical" effects to arise from "mundane" processes, or what have you.
10 - The weapon is possessed by a ghost, a demon, or some other spirit or ethereal creature. (Or perhaps an artificial intelligence if you want a little SF in your fantasy.)
11 - The weapon was imbued with supernatural significance due to a weird or particularly awful tragedy. Maybe it's the axe of an executioner who was tricked into beheading his own family. Maybe it's the first weapon to ever shed the blood of a child. Maybe it's the sword that was used to murder a benevolent empress and end the longest period of peace in history. Maybe it's the single arrow used over and over by a serial killer to shoot every one of his victims. Think of ghost stories or fairy tales or ancient mythology.**
12 - A god did it. Maybe they crafted or enchanted it on purpose. Maybe the weapon is just a figment of a god's imagination. Maybe the weapon is a piece of a god.***
*If a dungeon or other adventure location is overflowing with +1 weapons, I think this might serve as a better explanation for their presence than the idea that a magic-user cranked out a bunch of magic weapons that their class can't use, but your mileage may vary. I personally like this concept a lot, which is why it appears twice on the table, and why the table is so heavily weighted toward this result. You could kind of see this as a more specific form of Result 4, so if you want to adapt this table to other magic items besides weapons, it probably wouldn't hurt to switch Result 4 to Results 6 and 8, and come up with a new idea for the now-vacant Result 4.
**I got this idea from a plot device in the recent film Winchester. It wasn't a very good movie, unfortunately, but it did have some ideas that I think could be salvaged for better use in a different context. The movie's "magic weapon" of note came across as pretty darn silly (and unintentionally so, I believe), but I think it could have worked with a better set up, or in a different story.
***I guess this could overlap with just about any other result on the table. There's a lot of potential overlap between ideas on this table, really, but I think that's okay. My main purpose here was to brainstorm different origins for magic weapons, especially "common" ones, other than what I perceive as the standard "made by a wizard" explanation in fantasy RPG adventures.