Luckily it was a colleague who asked about the provenance of this word, not one of my students: for it was a word I had seen thousands of times and not investigated, in spite of my exhortations to them on such matters.

It is a word coined in the Greek of the New Testament era, and Latinized by Augustine and Jerome, on the assumption of a verb κατηχίζω, presumably a colloquialism.  But it was merely a synonym of κατήχησις—the oral teaching given to Christian converts before they were baptised. This latter noun is a firm derivative of the Classical word κατηχέω—to give forth a sound or resound in someone’s ears—and it is used in the New Testament. The verb is used in the passive also: and so people under instruction before baptism are called οἱ κατηχούμενοι—in English, catechumens.

Catechism in the sense of a printed statement of Christian doctrine in question-and-answer form seems to become current from the beginning of the 16th century, when John Colet insists that the pupils in the English schools then being founded across the realm should learn the catechism.

The general use of the word came about with the appearance of the Catechism in the Book of Common Prayer (1549).*  Shakespeare sometimes uses it jestingly, such as in As You Like It (Act III Scene 2), when Celia tells Rosalind that she has come across Orlando in the forest:

CELIA  Orlando.
ROSALIND   Alas the day, what shall I do with my doublet and hose! What did he when thou sawest him? What said he? How looked he? Wherein went he? What makes he here? Did he ask for me? Where remains he?  How parted he with thee? And when shalt thou see him again? Answer me in one word.
CELIA   You must borrow me Gargantua’s mouth first, ’tis a word too great for any mouth of this age’s size. To say ay and no to these particulars is more than to answer in a catechism.
ROSALIND   But doth he know that I am in this forest, and in man’s apparel? Looks he as freshly as he did the day he wrestled?
CELIA   It is as easy to count atomies as to resolve the propositions of a lover; but take a taste of my finding him, and relish it with good observance. I found him under a tree, like a dropped acorn—
ROSALIND   It may well be called Jove’s tree when it drops forth such fruit.
CELIA   Give me audience, good madam.

ἠχέω is a verb of choice for Homer: he and the tragedians use it of the roar of a crowd, or of a strong wind, or of the sea, or of singing or wailing,—in fact of any loud, dominating sound. And the adjective is memorable in Iliad I.157, when Achilles angrily reminds Agamemnon that he came to Troy for no personal grievance:

οὐ γὰρ πώποτ᾽ ἐμὰς βοῦς ἤλασαν οὐδὲ μὲν ἵππους,
οὐδέ ποτ᾽ ἐν Φθίῃ ἐριβώλακι βωτιανείρῃ
καρπὸν ἐδηλήσαντ᾽, ἐπεὶ ἦ μάλα πολλὰ μεταξὺ
οὔρεά τε σκιόεντα θάλασσά τε ἠχήεσσα:

Never did they rustle my oxen or my horses,
nor ever in deep-soiled Phthia, the nurse of men,
did they waste my harvest, since between is great space
of shadowy mountains and resounding sea…

Why Christians awaiting baptism or confirmation should be filled with sound is still not fully explained..

*Inspection recommended.