It means "sleeping heavily" or "sleeping soundly". I don't have the energy to do any research for you right now, but that phrasal verb might have been derived from a longer phrase like "to sleep the day away".
You two could be right. Macmillan says "sleep away" is a phrasal verb, but I've found nothing so far that specifically calls "snooze away" a phrasal verb. My guess is that "away" makes many different verbs act like phrasal verbs. It generally changes the meaning of "to X" into something like "to X heavily/continuously".
Note that "away" is sometimes used in imperatives as well.
1. (invented by me)
A married couple are conversing. "John, I have something important I want to talk to you about."
"I'm listening. Talk away!"
In this case it can be taken to mean "Go ahead. Talk as much as you wish!"
2. (traditional song) Way, haul away, we'll haul for better weather,
Way, haul away, we'll haul away Joe. http://www.contemplator.com/sea/hauljoe.html
In this case it is an injunction for the sailors to continue hauling until the sail is fully raised.
The OED contains this definition of the adverb away:
II.7 Onward in time, on, continuously, constantly; with idea of continuance of action and progress; e.g. to work away = to go on working.
Biffo's example of "Talk away!" uses the word in a slightly different sense, which is covered instead in the next definition:
II.8 Straightway, forthwith, directly, without hesitation or delay; chiefly colloquial in imperative sentences, as Fire away! = proceed at once to fire, begin immediately, Say away = say on, and U.S. and Eng. colloq. right away = straightway, directly.