As the word ἐλέγξει is not used in the New Testament only in this verse it is very helpful to look up its use in the LXX.
For example here, 'rebuke' is ἐλέγξει:
Let a righteous man strike me—it is a kindness; let him rebuke me—it is oil for my head; let my head not refuse it. (Psalms 141:5, ESV)
He who disciplines the nations, does he not rebuke? He who teaches man knowledge— (Psalms 94:10, ESV)
Or used slightly differently, 'judge' is ἐλέγξει:
And his delight shall be in the fear of the LORD. He shall not judge by what his eyes see, or decide disputes by what his ears hear, but with righteousness he shall judge the poor, and decide with equity for the meek of the earth; and he shall strike the earth with the rod of his mouth, and with the breath of his lips he shall kill the wicked. (Isaiah, Chapter 11:3-4, ESV)
Then again slightly different here, where 'reprove' is
Your evil will chastise you, and your apostasy will reprove you. Know and see that it is evil and bitter for you to forsake the LORD your God; the fear of me is not in you, declares the Lord GOD of hosts. (Jeremiah 2:19, ESV)
From these three contexts we see the word carries a varied sense. One is to show someone his fault and chide him for it with facts at the ready, i.e. rebuke. Another is to uncover the truth of a matter, expose it for what it is, and therefore decide on an outcome as a ‘judge’. The third is focusing on the evidence that brings about light on one’s own conscience, i.e. reprove and convince of the error.
Now in the case before us the translation into English is something as an operation of the ‘Spirit’ with respect to the ‘world’. The best translation must apply equally to ‘sin and righteousness and judgment’ and there is NO reason in any case (of sin, righteousness and judgment) that either rebuking, reproving or judging does not apply. Therefore:
- The Spirit rebukes the world in their sin for rejecting Him, as
well as uncovering the truth of the reality that it does so, as well
as bringing to light the perversity of doing so upon their conscience.
- Likewise the Spirit rebukes the world in their righteousness or lack of desire for it, as well as uncovering the truth of the reality of what it is, as well as bringing to light the nature of righteousness upon the conscience due to the fact that grace and righteousness is manifest in the resurrection.
- Likewise the Spirit judges the world in cursing it, as well as
uncovering the truth of the reality that it is judged, as well as
bringing to light the nature of judgment upon the conscience due to
the fact that Christ’s death has destroyed the Devil, the world's ruler, and his works of which the world is enslaved.
When we step back at the whole series of thoughts all tight joined up in this verse. It seems that we have to take both a positive and a negative sense of ἐλέγξει. This comes out with respect to righteousness for this is a positive subject. Also when God 'judges the poor' it is a positive ruling in their favor.
I think the footnote on this verse in JOHN PETER LANGE’s Commentary of John does justice to a proper conclusion:
Ver. 8.—[The E. V. reprove is certainly too weak for ἐλέξει, which implies both a convincing unto salvation and a convicting unto condemnation. See the EXEG. NOTES, and the remarks of Meyer (p. 551), who likewise ascribes to the ἔλεγξις the double aim of conversion (1 Cor. 14:24 f.) and condemnation (Acts 24:25; Rom. 11:7 ff.), in opposition to Erasmus, De Wette, etc., who confine it to the latter. The first example of the ἔλεγξις of the Spirit as effected through the apostles, is the pentecostal sermon of Peter, Acts 2 and its double effect. Webster and Wilkinson: “ἐλέγξει means (1) convince by proof, (2) convict, (3) reprove or rebuke … The passage is to be interpreted by the preaching of the apostles, or rather, of the Spirit by them (Matt. 10:20). In their discourses, recorded in the Acts, these three objects are the most prominent: (1) Christ the only Saviour, and rejection of Him fatal and damning sin. (2) Righteousness or justification, through the exaltation and intercession of Christ. (3) The kingdom of Christ, instead of Satan’s, now, and to be perfected in the final judgment.”—P. S.]
So which translation is better then?
- “Convict the world of guilt in regard to”: This seems valid but
does not seem to capture the positive sense of ἐλέξει.
- ”Will expose the guilt of the world”: Same as a)
- "Prove the world to be in the wrong about":
This is improved and better than a) or b). It is improved in that it does not insist 'guilt' i.e. consequential to the negative sense, and although it could indicate the positive sense more, at least it is close enough to the Greek meaning to allow both senses if one sees it in the context.