There should be two pulldowns. The first is the type of weapon, the second is the specific weapon that the trait will refer to. You need to buy a weapon of the type you pick in the first one, and then the second one will show it and you can pick it.
Also you must add a weapon to your character's gear before the second pulldown will populate. You can use the price adjustments while 'purchasing' it to do it... however Heirloom Weapon currently does it (it changed recently and I can't remember if you pay nothing or pay the base cost and get MW or what). Then you should be able to designate that weap as Heirloom.