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My 5e cleric needs to cast mending but it's not one of my memorized cantrips. Can he use one of his number of prepared spells to prepare a cantrip he doesn't know?

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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.

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    \$\begingroup\$ I actually just noticed that 5e switched the generic terminology to "known in mind", to cover both prepared and spontaneous casters. At least, that's what they use in the chapter on magic. \$\endgroup\$
    – starwed
    Oct 12, 2014 at 14:31
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    \$\begingroup\$ Note that it is still confusing since a cantrip is a spell and yet it does not use a slot or require preparation. \$\endgroup\$ Feb 22, 2015 at 22:58
  • \$\begingroup\$ @AlexisWilke I completely agree this can be confusing; they definitely buck the convention. \$\endgroup\$ Feb 22, 2015 at 23:08
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No. The cantrips you know are a fixed property of your character. Spellcasting classes all contain statements phrased as 'You know x cantrips' (where x is an integer). This implies that you know no more or less than x number of cantrips.

There are a number of ways to learn extra cantrips, these use statements phrased as 'You learn the [name of cantrip] cantrip' or 'You learn x cantrips'. If you knew all the cantrips but could only cast certain ones without preparing them, these statements should read something like 'You gain the ability to cast 'name of cantrip' at will, without preparing it or expending a spell slot.'

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    \$\begingroup\$ Exactly. You don't memorize cantrips, you know cantrips. You can't change what you know, at least, not easily. \$\endgroup\$
    – Adeptus
    Oct 9, 2014 at 4:08
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    \$\begingroup\$ It may be worth noting that a sufficiently high level cleric can directly petition their deity for aid, and "cast a cleric spell I dont have" is an excellent candidate for said aid. \$\endgroup\$
    – Bleep
    Oct 10, 2014 at 12:17
  • \$\begingroup\$ @DougM That doesn't seem relevant to the question. A high level Wizard or Sorcerer could use Wish to duplicate any cantrip, but the question was on whether you could prepare cantrips you don't know, not whether it was possible to use them. \$\endgroup\$
    – Miniman
    Oct 13, 2014 at 23:00
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    \$\begingroup\$ The question was specifically about a cleric wanting to cast mending. Alternate ways to gain the same effect are good answer components. (Isn't there at least one other cleric spell that can ape mending?) \$\endgroup\$
    – Bleep
    Oct 14, 2014 at 11:51
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RAW: No
RAI: This really ought to be possible

The cleric rules state that

At 1st level, you know three cantrips of your choice...

There is no process for swapping out cantrips. You know what you know, and can't put a new cantrip in one of your cantrip "slots".

It goes on to say...

You prepare the list of cleric spells that are available for you to cast, choosing from the cleric spell list.

There's no per-level limit on spells prepared, although...

the spells must be of a level for which you have spell slots

If you wanted to, you could prepare only 1st-level spells and cast them using higher-level slots. (This was technically true in 3.5/Pathfinder as well, but the spells didn't scale with the slot used (metamagic notwithstanding), so there was rarely a reason to do this.)

The only statement we see in RAW about cantrips and slots is that a cantrip "can be cast ... without using a spell slot". The rules don't say that "a cleric has a number of 0-level spell slots that are not expended when cast" like Pathfinder's orisons. Strictly by RAW, there are no 0-level slots. Since you can only prepare spells that correspond to your slots, you can't prepare 0-level spells.

So I draw either of two conclusions:

  1. The various restrictions to only the levels for which the character has slots are intended to apply to higher-level spells. The inability to prepare extra cantrips is an oversight on the part of the designers and should be addressed by houserules (and maybe a future erratum).
  2. The RAW function exactly as intended. Giving magic users the opportunity to sacrifice Burning Hands for Prestidigitation or Cure Wounds for Resistance was somehow undesirable to the designers and viewed as unbalancing or game-breaking.

Conclusion #1 sure makes a lot more sense to me. Prepared casters are meant to be versatile, and prohibiting extra cantrips cuts them off from some of the game's core utility spells.

I see two possible houserules/errata here:

  1. A character has 0th-level slots that are never expended when cast. All other issues with clerics preparing cantrips or wizards copying cantrips into spellbooks take care of themselves after that. The benefit of having an extra at-will spell is offset by the fact that the character has sacrificed the opportunity to prepare a higher-level spell.
  2. Change "of a level for which you have spell slots" to "of no higher level than that of your highest-level spell slot". A character may prepare or copy cantrips as other spells, but can only cast them using the slots explicitly granted in RAW.

Houserule #1 works more or less like Pathfinder and is more generous. From a mechanical standpoint, something like Thaumaturgy shouldn't cost the same as Inflict Wounds. From a flavor standpoint, it means that cantrips are cantrips because they're weak and don't expend any of the caster's energy when cast.

Houserule #2 is more in keeping with the flavor that 5e gives to cantrips: they are spells that are well-practiced by the caster and "fixed...in the caster's mind". Any other spell of similar power would require energy to cast because they haven't practiced it. Houserule #2 would work even better if cantrips could scale with spell level, but since they scale with caster level, there's no added benefit. And since you're expending a slot, you pay a steep cost for using that unmemorized cantrip.

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