In the films, Merry winds up with two swords (daggers, really, given hobbit scale). The first one Strider gives him on Weathertop. Strider could have gotten them from a barrow, but he shoulda said something. The other weapon that Merry has is the dagger that Galadrial gave him at the farewell scene in the extended edition. Galadrial's weapons have got to have enough mojo to take down the undead, that's for sure. She knows the power of rings, she is Noldor, kin to Feanor himself, and even Bilbo's Sting more than likely came from Thingol's armoury (some of which was made then looted by dwarves), where Galadrial herself was hanging out learning neat tricks from Melian.

But I agree - it's one of the holes in the films.

This is discussed over at the siteFox recomended, in their discussion of the move verses the book. It's over in the frame on the left, scroll on down.