"I have evidence that he plagiarized."
 
Michael Kirshner's answer about the above sentence is  good.  Michael says
that "it is an appositive in apposition to 'evidence.'"  Some grammar
reference books would claim just that.  Others would call the noun clause
"that he plagiarized" a complement to the noun "evidence."  It's not an odd
construction.  There are a lot of "factive" nouns that function together with
noun clauses in the same way.  The fact that..., the perception that..., etc.
 Notice that the nouns are all abstract nouns, usually related to an idea or
fact.  Often the nouns are derived from verbs, as in "He perceived that
George plagiarized," which would turn into "his perception that George
plagiarized."  Since the noun clause functions as a complent to the verb, it
likely is a complement to the nominalized verb as well.  Generative
grammarians also relate the factive nouns to the expletive  "it" in sentences
like
 
     It surprised me that he ate the pizza.
     The fact that he ate the pizza surpised me.
 
If "it" is used rather than a factive noun you have to extrapose the noun
clause, as in the first sentence above.
 
The main thing to see is that the NP+Clause makes a single constituent, a
noun phrase, with the noun clause functioning either as a complement or an
appositive.   I'm not sure the term you use (complement or appositive) is
that important.   It is important to see that the clause is a noun clause,
not a relative clause and that the "that" is a subordinate conjunction.
Cheers, Max Morenberg