This is an invention of Peter Jackson & co and has absolutely nothing to do with the books, so the only relevant sources we can use for answering the question are the movies.
However, the last 2 Hobbit movies have yet to be released.
So it's going to be a clear "WAFO" ("Watch and find out") answer to this one.
Update - 16th November 2013
In the commentary for the Extended Edition of AUJ, Peter Jackson confirms that the sword Radagast finds is the same sword that the Witch King used to stab Frodo, and Philippa Boyens remarks:
It comes into play even more. There's a reason for it.
Finally Jackson comments:
A lot of this is very early setup material for events which are expanded on in the next two Hobbit movies.
This just serves to underline the fact that it's a "watch and find out" answer. You shouldn't expect to be given all information up-front, particularly with a 3 part movie series of which the last two parts have yet to be released.
Update - 15th December 2013
No, there was nothing on it in the second movie, unless there are extra scenes relating to it being held back for the extended edition. It looks possible that the Nazgul are going to put in an appearance in the third movie, however, so either way we must continue waiting.
Update - 2nd July 2014
According to this page at TheOneRing.net it's actually Elrond who gives it back to him! We're apparently going to see this happen in the third movie. Scroll down to the section headed "Elrond and the Morgul-blade" and we read:
Ringer LaurelinDena reminds of us of this snippet from the Weta Book 'Cloaks and Daggers' on page 166 where swordmaster Peter Lyon talks about the Morgul Blade.
Later we came to film a scene in which Elrond is seen casting the blade into the Witch-King’s tomb...
I don't have access to the cited source and so can't confirm the authenticity of this quote, but for now it's the best explanation we have. Unless the Desolation of Smaug Extended Edition upsets things, we'll be able to confirm come December.
Update - 1st November 2014
No, there was nothing about it in the Extended Edition of DoS either, although there was a scene of the supposed original burial of the Witch-king, with his sword being thrown into his tomb, and I'm wondering if this is the scene that was referred to above (I'll need to rewatch it to check if it was Elrond). I haven't yet listened to the commentary though.
So unless the commentary reveals any details we're definitely going to have to wait on the third movie.
(7 hours later: I've rewatched and listened to the commentary on that scene, and it's Men who are burying the WK, with Royd Tolkien (JRRT's great-grandson) playing the (unnamed) character throwing in the sword. So that confirms that it's not the same scene.)
Update - 14th December 2014
Nothing about it in the third movie; I guess we'll have to wait for the extended edition.