Do daily spells (spells that include "once per day" the description) recharge after 24 hours, or could they be cast at 23:59 (just before midnight), and cast again at 00:01 (just after midnight)?
The answer is neither.
Info gleaned from the section on Magic in the PRD.
Spells and Daily items have a recharge limit
Recent Casting Limit: As with arcane spells, at the time of preparation any spells cast within the previous 8 hours counts against the number of spells that can be prepared.
So any spell cast recently counts against the number that can be prepared, and for daily spell-like effects this means once used they cannot be reused (effectively their 'slot' is used) until 8 hours later. This is why casters usually get a night's sleep before preparing their spells!
But why are daily items treated like spells?
How do daily items recharge?
Daily spell-like abilities recharge whenever the caster would normally be able to relearn spells as noted at the bottom of the Magic section by Special Abilities:
Spell-Like Abilities: Usually, a spell-like ability works just like the spell of that name. A spell-like ability has no verbal, somatic, or material component, nor does it require a focus. The user activates it mentally. Armour never affects a spell-like ability's use, even if the ability resembles an arcane spell with a somatic component.
A spell-like ability has a casting time of 1 standard action unless noted otherwise in the ability or spell description. In all other ways, a spell-like ability functions just like a spell.
Therefore daily spells will be recovered by the caster whenever they would normally recover a spell and daily items are limited by the recharge limit as well.
Relearning Spells (For completeness)
For arcane spellcasters this would be after an 8 hour period of rest, for divine spellcasters this would be at the set time that they recover spells. For non-spellcasters that have a spell as a special ability I would require them to complete a period of rest to regain their spells.
Added to this the caster needs a small amount of time to prepare their spells.
Wizards Require an hour, or a minimum of 15 minutes to add/remove other spells (proportionally lower)
Sorcerers/Bards Require 15 minutes of concentration to update their spells.
Divine casters have the same praying preparation time as Wizards (1 hour or 15 minutes reduced)
There is a wizard feat to prepare spells faster, I'm sure, but I can't find it at the moment...