With all due respect to Mr. Koukl, Biblical repentance (the kind we preach when preach Jesus' gospel of repentance and faith) has to do directly with sin. Repentance is a turning from sin to Christ. The Westminster Shorter Catechism puts it well:
Q. 87. What is repentance unto life?
A. Repentance unto life is a saving grace, whereby a sinner, out of a true sense of his sin, and apprehension of the mercy of God in Christ, doth, with grief and hatred of his sin, turn from it unto God, with full purpose of, and endeavor after, new obedience.
The Westminster Larger Catechism has a similar description:
Q. 76. What is repentance unto life?
A. Repentance unto life is a saving grace, wrought in the heart of a sinner by the Spirit and Word of God, whereby, out of the sight and sense, not only of the danger, but also of the filthiness and odiousness of his sins, and upon the apprehension of God’s mercy in Christ to such as are penitent, he so grieves for and hates his sins, as that he turns from them all to God, purposing and endeavouring constantly to walk with him in all the ways of new obedience.
My Reformed Baptist brethen may enjoy the London Baptist Confession of Faith (1689):
OF REPENTANCE UNTO LIFE AND SALVATION
Paragraph 1. Such of the elect that are converted at riper years, having sometime lived in the state of nature, and therein served divers pleasures, God in their effectual calling gives them repentance to life.
Paragraph 2. Whereas there is none that does good and does not sin, and the best of men may, through the power and deceitfulness of their corruption dwelling in them, with the prevalency of temptation, fall in to great sins and provocations; God has, in the covenant of grace, mercifully provided that believers so sinning and falling be renewed through repentance unto salvation.
Paragraph 3. This saving repentance is an evangelical grace, whereby a person, being by the Holy Spirit made sensible of the manifold evils of his sin, does, by faith in Christ, humble himself for it with godly sorrow, detestation of it, and self-abhorrancy, praying for pardon and strength of grace, with a purpose and endeavor, by supplies of the Spirit, to walk before God unto all well-pleasing in all things.
Paragraph 4. As repentance is to be continued through the whole course of our lives, upon the account of the body of death, and the motions thereof, so it is every man’s duty to repent of his particular known sins particularly.
Paragraph 5. Such is the provision which God has made through Christ in the covenant of grace for the preservation of believers unto salvation, that although there is no sin so small but it deserves damnation, yet there is no sin so great that it shall bring damnation to them that repent, which makes the constant preaching of repentance necessary.
Mr. Koukl makes hay of the fact that the Greek word for repentance, μετανοέω (metanoeō), does not necessarily have to do with sin. That's absolutely true: it means a change of mind. The word itself doesn't even necessarily have to do with God. Like the English word "turn" it can be applied to various things. Thus, we sometimes see it used in Scripture of things other than turning to God (God even anthropomorphically describes himself as repenting from some thing he would otherwise have done).
Those etymological and linguistic arguments miss the mark. Godly repentance, the kind that Christ preached, is two sided: it is a turning from sin, and a turning to God. It would be wrong to preach a one-sided repentance that only addressed sin, but it is also wrong to preach a one-sided repentance that does not address sin.
What's more, while repentance is toward God, that's normally only implied in Scripture, whereas repentance from sin (despite Koukl's claimed word study) is frequently expressly or indirectly stated.
In the Old Testament, we see a few examples:
Jeremiah 8:6 I hearkened and heard, but they spake not aright: no man repented him of his wickedness, saying, What have I done? every one turned to his course, as the horse rusheth into the battle.
Ezekiel 14:6 Therefore say unto the house of Israel, Thus saith the Lord GOD; Repent, and turn yourselves from your idols; and turn away your faces from all your abominations.
Ezekiel 18:30 Therefore I will judge you, O house of Israel, every one according to his ways, saith the Lord GOD. Repent, and turn yourselves from all your transgressions; so iniquity shall not be your ruin.
We also see additional examples, that I won't provide here of the usage of the term "repentance" for other things than turning from sin. Nevertheless, it should be clear that the emphasis in those cases is normally on the negation of some prior course of action or behavior. God "repents" of his plan to destroy Israel, or something like that. I welcome folks to try to prove me wrong, but it seems like the emphasis of repentance throughout Scripture is on the negative - the turning from, rather than the turning toward (which is normally much more indirectly indicated).
We are not limited to the Old Testament, of course, and the New Testament provides even more examples of the same connection between sin and repentance:
Matthew 9:13 But go ye and learn what that meaneth, I will have mercy, and not sacrifice: for I am not come to call the righteous, but sinners to repentance.
Mark 1:4 John did baptize in the wilderness, and preach the baptism of repentance for the remission of sins.
Mark 2:17 When Jesus heard it, he saith unto them, They that are whole have no need of the physician, but they that are sick: I came not to call the righteous, but sinners to repentance.
Luke 3:3 And he came into all the country about Jordan, preaching the baptism of repentance for the remission of sins;
Luke 5:32 I came not to call the righteous, but sinners to repentance.
Luke 15:10 Likewise, I say unto you, there is joy in the presence of the angels of God over one sinner that repenteth.
Take heed to yourselves: If thy brother trespass against thee, rebuke him; and if he repent, forgive him. And if he trespass against thee seven times in a day, and seven times in a day turn again to thee, saying, I repent; thou shalt forgive him.
Luke 24:47 And that repentance and remission of sins should be preached in his name among all nations, beginning at Jerusalem.
Acts 2:38 Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.
Acts 3:19 Repent ye therefore, and be converted, that your sins may be blotted out, when the times of refreshing shall come from the presence of the Lord;
Acts 5:31 Him hath God exalted with his right hand to be a Prince and a Saviour, for to give repentance to Israel, and forgiveness of sins.
Acts 8:22 Repent therefore of this thy wickedness, and pray God, if perhaps the thought of thine heart may be forgiven thee.
Acts 26:20 But shewed first unto them of Damascus, and at Jerusalem, and throughout all the coasts of Judaea, and then to the Gentiles, that they should repent and turn to God, and do works meet for repentance.
2 Corinthians 12:21 And lest, when I come again, my God will humble me among you, and that I shall bewail many which have sinned already, and have not repented of the uncleanness and fornication and lasciviousness which they have committed.
Hebrews 6:1 Therefore leaving the principles of the doctrine of Christ, let us go on unto perfection; not laying again the foundation of repentance from dead works, and of faith toward God,
And I gave her space to repent of her fornication; and she repented not. Behold, I will cast her into a bed, and them that commit adultery with her into great tribulation, except they repent of their deeds.
And the rest of the men which were not killed by these plagues yet repented not of the works of their hands, that they should not worship devils, and idols of gold, and silver, and brass, and stone, and of wood: which neither can see, nor hear, nor walk: neither repented they of their murders, nor of their sorceries, nor of their fornication, nor of their thefts.
I've omitted the cases where the connection was less obvious and immediate, and so I hope that my readers will find the above examples sufficient to illustrate the matter. Repentance of the Biblical variety has much to do with sin, in fact it is a turning from sin that we preach when we tell the lost to "Repent and Believe" as Jesus our Master instructed.