Well. I guess the question still remains; why not just say "the origin of craving should be abandoned"? Is this the same as saying "the truth of the origin of suffering should be abandoned"? Or is this somehow different?
Well, the origin of suffering, according to Buddhism, is "craving". So, when one says that the truth of the origin of suffering is craving, one is establishing the fact that a truth exists about the origin of suffering as well as, in a codependent way, enunciating the content of that truth.
So, it seems the problem here is that, in accordance with common sense, people don't normally abandon truth; especially if those "truths" are your guide towards some ultimate goal. But, in this case, when you abandon the truth of the origin of suffering, you simply abandon what that truth contains.
I think the misunderstanding arises from this. You read; "The Truth of the Origin of Suffering" -should be abandoned. This implies that any knowledge anyone might acquire from such a truth is to be thrown away; which simply doesn't make any sense. But, actually, the proper way to read the statement is; the "truth" of the origin of suffering should be abandoned. And because that "truth" is craving, the truth of the origin of suffering and the origin of suffering do mean the same thing, as someone said long ago up above.
The "Truth of the Origin of Suffering", as far as we take it only to mean "knowledge" should (probably) not be abandoned. In other words, it is (likely) good to keep in mind, as much as you can, that craving is the origin of suffering.
The "truth" of the origin of suffering, in as much as we recognize or penetrate into the realization that craving, and craving alone, is the true origin of all suffering; -this we should certainly abandon. So, emphasis on the word "truth".
Also Chapter 16 in Buddhaghosa's book gives a very good explanation about how the four noble truths are understood in a mundane and supermundane way. That chapter might also better explain or give an answer to the Oppy's question.