The word ἐπερώτημα (eperōtēma) should probably really be translated "examination" or "inquiry" and not "request". It is used in this sense (in verb form - ἐπερωτάω) in Romans:
Ἡσαΐας δὲ ἀποτολμᾷ καὶ λέγει· εὑρέθην τοῖς ἐμὲ μὴ ζητοῦσιν, ἐμφανὴς ἐγενόμην τοῖς ἐμὲ μὴ ἐπερωτῶσι.
Then Isaiah is so bold as to say, “I have been found by those who did not seek me; I have shown myself to those who did not inquire after me” (Romans 10:20)
where Paul is quoting verbatim from the Greek Septuagint version of Isaiah 65:1.
A footnote in The Orthodox New Testament (not to be confused with The Orthodox Study Bible) notes that ἐπερωτάω and ἐπερώτημα were "used in ancient times with the sense of approval or sanction after inquiry of a higher authority as the senate or Areopagos."1
The Greek phrase συνειδήσεως ἐπερώτημα literally means an "examination of conscience" and means exactly the same thing as the English phrase. The ONT translates the verse:
There is also an antitype which now saveth us - baptism (not a putting away of the filth of the flesh, but the examination of a good conscience toward God.)
This is also the way that Greek Church Fathers understood the passage,2 but only Young's Literal Translation (cited in another answer) seems to come close to this.
This examination of conscience as it relates to baptism is perhaps brought to mind in Peter's instruction at Pentecost: Repent, and be baptized (Acts 2:38). Repentance - μετάνοια (metanoia) - is a changing of one's spiritual being (νους - nous), the Greek word meaning not merely contrition or regret, but "more positively and fundamentally the conversion or turning of our whole life towards God."2
With the above in mind, Bede (672-735) explained the verse:
... Not the removal fo the body's dirt but as the examination of a good conscience for God
For where is a good conscience except where there is sincere faith? For the apostle Paul teaches that the purpose of the commandment is charity [love] from a pure heart and a good conscience and unfeigned faith (1 Timothy 1:5). The fact, therefore, that the water of the flood did not save those outside the ark but slew them without doubt prefigured every heretic who, although having the sacrament of baptism, is to be plunged into the outer waters by which the ark is raised up to the heavens.3
1. The Orthodox New Testament, Vol 2., Acts, Epistles, and Revelation (Holy Apostles Convent, 2000), p.463.
2. The Philokalia, Vol. 1, (tr. from Greek; Faber and Faber, 1979), p.364
3. Commentary on the Seven Catholic Epistles (tr. from Latin, Cistercian Publications; 1985), p.105
