Depends on the weapon's classification (Blue, Green, White, Pink, etc.), among other factors.

Here are some things I do with outdated weapons. Out of those, none of them is sell them to an NPC. Note* donations to Doman Enclave does not count since I get double the rate.

* Desynth - This is basically the last resort; or rather, the weapon has no value whatsoever outside what is returned via RNG. If I couldn't desynth, or the mats are undesired; then I would donate it to the Doman Enclave.
* Trade for GC seals - Only applies to pinks, greens, and blues.
* Place on the MB - Only applies to whites and certain greens at 0% SB. This is usually what happens with what retainers bring back and mob drops.
* Placed into the glamour dresser - What every weapon the WoL uses desires to be. To be given the honor of glamouring over my ugly arse BiS tome weapon is highly commendable indeed. ^^