I believe what she said. I believe that she said it.
    To me, the fact that "that she said it" can fit in the same slot as "what she said" is strong motivation for declaring it a direct object. I know "that" is a complementizer (and an optional element), but I'm not sure why the boundary for what gets accepted as direct object can't be wide enough to include it. 
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Regardless of the approaches we take, we have to have some kind of terminology:

"I" serves as a subject, a pronoun, an NP

"Believe" is a verb

"She said what" and "she said it" are clauses that are complements of the verb "believe" (VP)

In the second case "that" serves only as a word that introduces the noun clause

Yes, of course Direct Object is a meaningful term.
Gerald

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