Technically though, the DMG (p. 40) gives actual amounts of rest required to be able to re-memorize spells of any given level:
Strictly speaking, you don't need a full night's rest to re-memorize your spells if they're of a low enough level.
Memorization time is also laid out on the same page: 15 minutes, per spell, per spell level. So 15 minutes to memorize a first level spell, 30 minutes to memorize two first level spells, 60 minutes to memorize two first level and one second level spell, etc.
I'm really not sure to what extent these rules have an effect on play. If I were to speculate (I'm not asserting this to be the case by any means), it might become an issue at higher levels. At level 12, for example, it would take 8 hours of sleep and 16.5 hours of memorization to replenish a magic-user's entire allotment of spells – a total of 24.5 hours of spell-recovery time. So there is a theoretical break point of sorts where an entire day would be insufficient to replenish a full allotment of spells.
But of course to need that much recovery time, in practice, you'd have to cast 21 spells in one day. Assuming the caster sleeps for 8 hours, that comes to more than one spell per hour. I suppose there could be significant effects in some sorts of extended combat- and obstacle-dense adventures, where the party is pressed for time and/or rest gets interrupted easily.
At any rate, it's a rule I'd like to see in practice just to know what effect it has, if any.