Elihu continues the accusations of Eliphaz, Bildad, and Zophar, lacks their wisdom, and is beneath contempt.
Structure of the dialogue
The core of Job follows a strict order of speeches for three cycles:
The basic content of Job's words is always the same: questioning God why the calamities of chapter 1 and 2 came upon him. We know from those chapters that Job was singled out for being "blameless and upright man who fears God and shuns evil". But his friend's speeches try to convince Job of the precise opposite: he is being punished for sin. As readers, we are signaled to sympathize with Job, who really receives unfair treatment at the hands of God, the Adversary, and his friends.
In his final speech in the the final cycle, Job demands an audience before God:
O that I had someone to give me a hearing;
O that Shaddai would reply to my writ,
Or my accuser draw up a true bill!—Job 31:35 (NJPS)
From chapter 27 to 31, Job swears what amounts to an oath of innocence. The concept stems from the Babylonian legal system in which an accused could clear themselves by swearing they are innocent. The effect of such oaths was to compel an accuser to bring evidence against the accused. If the accuser did not have evidence, the accused was declared innocent and could press false witness charges to his accuser.
It seems the three friends took this as the end of the dialogue, since it puts the onus on God to prove the charge that Job sinned:
These three men ceased replying to Job, for he considered himself right.—Job 32:1 (NJPS)
The structure of the book is broken at this point, since Zophar can not continue to charge Job with sin against God—Job has taken the proper legal step to answer such a charge. At this point, only God has legal standing and if He does not speak, Job has reason to charge Him with false punishment. So when Elihu stands up, he is simultaneously completing the pattern and speaking out of turn. Structurally, despite his protestations, Elihu stands on the side of Job's accusers.
Elihu the fool
When the author of Job introduces Elihu, it is not particularly flattering:
Then Elihu son of Barachel the Buzite, of the family of Ram, was angry—angry at Job because he thought himself right against God. He was angry as well at his three friends, because they found no reply, but merely condemned Job. Elihu waited out Job’s speech, for they were all older than he. But when Elihu saw that the three men had nothing to reply, he was angry.—Job 32:2-5 (NJPS)
I've italicized three terms that carry negative connotation:
Buzite—literally, "contempt". The other names associated with Elihu (meaning "He is my God") are not negative, so this is not decisive. (Barachel means "God blesses" and Ram means "high" or "exalted".) It should be noted that while Job's name (meaning "hated") seems symbolic, none of the other friend's names seem to be particularly helpful in understanding their character.
Angry—carries a connotation of what we might call in English "flaring nostrils" or "red in the face". Anger was not a valued trait according to ancient wisdom. (See Proverbs 29:8) Worse, there is some indication he was drunk:
Now I also would have my say;
I too would like to hold forth,
For I am full of words;
The wind in my belly presses me.
My belly is like wine not yet opened,
Like jugs of new wine ready to burst.
Let me speak, then, and get relief;
Let me open my lips and reply.
I would not show regard for any man,
Or temper my speech for anyone’s sake;
For I do not know how to temper my speech—
My Maker would soon carry me off!—Job 32:17-22 (NJPS)
By his own words, Elihu seems a man out of control. It's possible to read this as "righteous anger", but I don't see support of this idea in the text.
Youth—specifically, he labels himself as being "small in days". Being younger than the others is Elihu's reason for not speaking up earlier, but the ancient custom was that age was associated with wisdom. Elihu defies that tradition:
It is not the aged who are wise,
The elders, who understand how to judge.—Job 32:9 (NJPS)
While our culture easily accepts these words, it would have been an almost insurmountable obstacle to the original readers of Job.
Combining these indicators together, we get a strong sense that Elihu is set up to be a fool.
Elihu still accuses Job of sin
Despite claiming that he won't repeat the friends' arguments (Job 32:14), Elihu persists in accusing Job of wrongdoing:
Would that Job were tried to the limit
For answers which befit sinful men.
He adds to his sin;
He increases his transgression among us;
He multiplies his statements against God.—Job 34:36-37 (NJPS)
His primary argument, that God uses pain and suffering to discipline the righteous has already been broached by Eliphaz in chapter 5.
Elihu is wrong about God
Further, Elihu says that God does not need to respond to Job's case (or that He can do so in His own time depending on the translation of Job 35:12-16). He rhetorical asks what effect sin and righteousness have on God:
If you sin, what do you do to Him?
If your transgressions are many,
How do you affect Him?
If you are righteous,
What do you give Him;
What does He receive from your hand?—Job 35:6-7 (NJPS)
The implied answer to each is a variation on "nothing". In what must be the most devastating courtroom tactic every employed, God blows into the scene (literally) and asks "Who is this who darkens counsel, speaking without knowledge?" (Job 38:2) While the text does say that God responds to Job, He also is undercutting Elihu's argument.
Why doesn't God rebuke Elihu?
Perhaps the most powerful argument for Elihu is God's silence about his argument. The other three friends are rebuked and required to make sacrifice in the epilogue. Job himself is commended for all that he had done and receives a double blessing. But Elihu disappears from the story. If God were displeased with Elihu, why didn't He include him in the rebuke?
On the other hand, why didn't God praise Elihu? The book of Job purposely leaves his status ambiguous: he could be correct in his argument or he could be incorrect. But a close reading of Elihu's words (as I show above) reveals that he simply amplifies the other (incorrect) theodicies or distances God from caring about humanity. Further, he speaks out of turn and has no standing before God. Like a man who disrupts the order of the court, he is quickly and quietly dismissed. Elihu is not even worthy of being answered.
The bulk of this argument comes from Putting God on Trial: The Biblical Book of Job by Robert Sutherland.
The "Elihu" entry in the Eerdmans Dictionary of the Bible provided some further hints.