μετανοέω (μετανοῶ) is composed of the preposition μετά and the verb νοέω. The preposition μετά in composition may indicate “change of place, condition, plan, etc.”1 The verb νοέω means, among other things, “to understand; to think.”2 Thus, μετανοέω means, among other things,3 “to change how or what one thinks.”4

In English, we also say, “to change one’s mind”:5

In Cyropaedia, Xenophon wrote,6

In Epitrepontes (Επιτρέποντες), Menander wrote,7

In addition to meaning “to change one’s mind,” LSJ also defines it as “repent.”8

“Repent” is defined by the Oxford English Dictionary as:9

Now, “repent” (and μετανοῶ, 3.) still implies a change of mind, but this change is caused by one’s own contrition, regret, or being sorry for one’s own action. In μετανοῶ, 2., the change of mind is not necessarily caused by such regret. Rather, it could simply be a matter of convenience.
Many assume that this meaning of μετανοῶ (i.e., 3.) was born by the Greek New Testament. However, already in the 5th c. B.C., Antiphon of Rhamnus had used the verb μετανοῶ in the same sense.10
ταῦτα οὖν σεβόμενοι ὁσίως καὶ δικαίως ἀπολύετέ με, καὶ μὴ μετανοήσαντες τὴν ἁμαρτίαν γνῶτε: ἀνίατος γὰρ ἡ μετάνοια τῶν τοιούτων ἐστίν.
Respect these considerations, and satisfy heaven and justice by acquitting me. Do not wait until remorse proves to you your mistake; remorse in cases such as this has no remedy.
Maidment translates both the verb (participle) μετανοήσαντες and the noun μετάνοια11 by a form of “remorse.” This is clearly an appropriate English translation consdering the context. It is remorse or regret for one’s action that causes the individual to change his mind, i.e., repent.
In summary, μετανοῶ is appropriately translated as “repent” in English translations of Acts 2:38, because it does not simply mean “to change one’s mind,” but to change one’s mind because of the contrition, grief, remorse, or guilt caused by the consciousness of one’s sins. Hence, the apostle Paul writes that the Corinthians “were grieved unto repentance,”12 and “godly sorrow produces repentance unto salvation.”13
Footnotes
1 LSJ, p. 1109, μετά, G., VIII.
2 id., p. 1177, νοέω, 2. & 3.
3 LSJ, p. 1115, μετανοέω, lists its first meaning as “perceive afterwards or too late; concur subsequently.”
4 OED online, mind (n.), 12., “to change one’s mind”
5
6 Book 1, Ch. 1, §3
7 Act 2, Lines 70–75.
8 LSJ, p. 1115, μετανοέω, 3.
9 OED online, repent (v.), 1., a.
10 First Tetralogy, 2.4.12; also, see On the Murder of Herodes, 5.91
11 “repentance”
12 2 Cor. 7:9
13 2 Cor. 7:10
References
Liddell, Henry George; Scott, Robert; et al. A Greek-English Lexicon. 9th ed. with revised supplement. Oxford: Clarendon, 1996.
Menander. Four Plays of Menander: The Hero, Epitrepontes, Periceiromene and Samia. Ed. Capps, Edward. Boston: Ginn, 1910.
Menander. The Arbitration (The Epitrepontes of Menander). Trans. Murray, George. London: George Allen & Unwin, 1945.
Minor Attic Orators, Volume I: Antiphon. Andocides. Trans. Maidment, K. J. Cambridge: Harvard UP, 1941.
Oxford English Dictionary online.
Xenophon. Xenophontis Opera Omnia. Reprint. 1970. Oxford: Clarendon, 1910.
Xenophon. Cyropaedia, Books 1–4. Trans. Miller, Walter. Vol. 1. Cambridge: Harvard UP, 1914.