I am working on a different approach - the crafting rules in Xanathar's indicate that special ingredients or materials are required for making magic items - so, I simply have it that there's no difference between "magic weapon" and "weapon made from special materials".
This gets around some of the weirdness in which, technically, folks not proficient in Arcana can make magic items - for these, the above-normal abilities come from the material and details of forging process, not necessarily casting enchantments they'd not know how to use.
Regardless of edition, this is my preferred way to handle such things. IMO (and somewhat IME) it makes the world feel more magical--because the "riddle of steel" and otherwise being a smith of peerless quality really DOES mean you can have things transcend the limits of mundanity.
There are also materials lost to us now that really did exist, like "hepatizon" (literally "liver-colored") which was a special and extremely valuable type of bronze, or Damascus steel. And other real-world mythical materials that may or may not have existed, like "flexible glass," which could
allegedly be beaten like a ductile metal without breaking.
Materials I've suggested elsewhere:
ravenglass: obsidian specifically formed by conflicting magical energy, often "water elemental quenching lava" or "casting blizzard and meteor storm in the same area." Develops a similar absurdly-sharp edge as regular obsidian, but much less prone to breakage. Not infinitely durable, but durable enough to make solid pieces out of (think "ebony" and/or "glass" weapons from Elder Scrolls).
vertigis: unusual green-hued metal that affects the "flow" of magic around it in different ways depending on what it's alloyed with. Vertisteel acts as a magic "sink," "grounding" magical fields around it (improving defense against magic but also making it harder to benefit from positive magic). Mythigis (mythril/vertigis alloy), on the other hand, acts as an overall magic amplifier, enhancing both positive and negative magic applied to the wearer.
unmelting ice: lightweight and obviously cold, but requires some minimal maintenance (e.g. water-polishing it to fill in nicks and dents), basically impossible to forge but can be slowly "grown" from seed crystals under the right conditions (no ice-nine here).