Phrasing "An hour's rest"

This is what the Chicago Manual of Style terms an "idiomatic shorthand form of an of-phrase:"

an hour's delay is equivalent to a delay of one hour, or a one-hour delay

So: an hour's rest can be phrased as a one-hour rest. (I find a rest of one hour a bit stilted...)

Edit: As noted in my comment, in this idiom, delay or rest "belongs" to hour (it's of one hour), so it's "an hour's delay."

I've always seen it with the apostrophe. Without would look goofy. Wikipedia has an explanation as to why (not that it's a standard grammar reference but I believe in this case it's correct):

An apostrophe is used in time and money references, among others, in constructions such as one hour's respite, two weeks' holiday, a dollar's worth, five pounds' worth, one mile's drive from here. This is like an ordinary possessive use. For example, one hour's respite means a respite of one hour (exactly as the cat's whiskers means the whiskers of the cat). Exceptions are accounted for in the same way: three months pregnant (in modern usage, we do not say pregnant of three months, nor one month(')s pregnant).

No one is going to object if you write an hour's rest. It's often misleading to think of the apostrophe s as expressing possession, at least in the sense in which it is commonly understood. It makes more sense to think of it as a remnant of the genitive case, capable of expressing a range of meanings. However, there are signs that, in such cases as this, the apostrophe mark itself is disappearing and, as far as I'm concerned, good riddance.

Yep, the delay belongs to the hour. Kinda weird when you think about an hour owning something, but it's the noun-adjectival link which forces phrases like this into the rule.

Besides, think of it like this. How can you ever have "an hours rest'? One hours? The rest cannot be modified by "an hours" because it's a glaring grammatical error! It is modified by "an hour", and so that's why the apostrophe is needed.