No, you can't prepare Cantrips. They're by definition non-prepared, or rather, they're spells you simply know and have permanently prepared without doing anything:

Cantrips

A cantrip is a spell that can be cast at will, without using a spell slot and without being prepared in advance. Repeated practice has fixed the spell in the caster’s mind and infused the caster with the magic needed to produce the effect over and over. A cantrip’s spell level is 0.

It's worth distinguishing that spells don't get memorised as you said — they get prepared*, which is apparently a more involved process than remembering some words and motions. In the case of cantrips, whatever preparation gives you, the caster now has permanently on hand.

Flavour-wise it sounds like cantrips might be the sort of thing you still should be able to prepare. However, you'd need 0-level spell slots to be able to prepare a cantrip. Nobody has those. From the Cleric section on preparing spells:

You prepare the list of cleric spells that are available for you to cast, choosing from the cleric spell list. When you do so, choose a number of cleric spells equal to your Wisdom modifier + your cleric level (minimum of one spell). The spells must be of a level for which you have spell slots.

The case appears pretty watertight. In fact, the author of the Cantrips section seems like they expected Cantrips to never get prepared at all, since that language reads really strangely if you somehow assume preparing cantrips is something you can do.

It seems from your answer that you're surprised they didn't allow people to flexibly pick out extra cantrips, and think they should do it differently — but that's another matter altogether. They appear to have designed it this way deliberately and without accident.

^{* The idea that spells get memorised is a misnomer that appeared somewhere in D&D's history. You can read about what that means here, though the narrative logic might be different in 5e.}