ToME Wiki
Finding and recognizing Narsil
There is no one likely location that Narsil can be found in.
To help you recognize Narsil if you find it:
Destroy (or ID) every broken sword you find. If you happen across one that cannot be destroyed, keep it!, for you may have found the famed Narsil.
- Alternatively, you can check the damage dice of every broken sword you find. Narsil has 3d2 damage, whereas normal broken swords only have 1d2.
However, you must ID Narsil in order to receive the quest to have it be reforged.
Reforging Narsil
If it really is Narsil, take it to Aragorn in Minas Anor. He will reforge it into Anduril and then give it to you as a reward.
Chatter
MayLith: I'm not sure whether Aragorn will accept Narsil if it is unidentified. Never tried it.
ErisDiscordia: I found Narsil recently and got a special ("yellow") message very shortly after finding it. Is that only upon ID'ing it? Either way, that should probably be mentioned here.
MayLith: I don't remember -- I've only found Narsil once and that was a while ago. Ummm.../me looks cross-eyed at the code... q_narsil.c only has Aragorn's speech in it, not the blurb when you first find it, and I don't know where else to look.
BTW, is it just me, or could Aragorn's speech be reworked a little?? 
Anyway, I agree with you.
MassimilianoMarangio: As for most quests, the code is in q_foo.c and the quest text in tables.c:
MayLith: (me the one-day-old non-veteran of diving ToME code: Gee, that's logical!
Thanks for telling me though.)
You have found Narsil, a broken sword. It is said that the sword that was broken shall be reforged... Maybe it is this one. You should bring it to Aragorn at Minas Anor -- he would know.
Looking at the code, you can only return Narsil to Aragon if the quest was taken, i.e. if Narsil was identified.