Ok, so a +2 weapon is pretty good but a bit bland so...
the DMG has the answer! Look at the "special features" - some of them are already chosen - like the creators/origins are dwarves, for example, but others you can roll randomly. Maybe it's a "signature" of the smith. Maybe it will be related to the special ingredient.
... and that should point you towards what the weapon is made of.
I'll give you an example for a +1 battleaxe:
"THE AXE
shark tooth
Gurthlimbnél (literally, death fish tooth in sindarin)
The haft is made of wood from the Grandfather Tree. My clan are its guardian. From time to time the Tree sheds branches. The wood is collected and used as it has magical properties, especially for life magic. In this case, the strong living essence within the mood nourishes the tooth, allowing it to constantly repair itself and remain ever sharp.
The haft was given to a group of sea-elves, who found the mastodon tooth and bound it to it, creating Sharktooth.
About 200 years ago, the axe was stolen by a pirate, who wielded it in a bloody 15 year career and earned himself the name "SharkTooth"
However, his ship eventually sank and the axe was lost. Now it is back in elven hands. If I were to perish in this endeavor, please return the axe to elven hands, to my clan preferably - my time outside the sacred woulds will have been very short, but my finding of the axe will be worthy of a song."
This was based on creators: elves, heroic, waterborne and hungry. We modified it slightly to allow it to be wielded underwater without disadvantage.