Snooze is of uncertain origin according to the Oxford English Dictionary, "snooze, v.":
apparently a cant or slang word of obscure origin.
The earliest usages establish a general meaning but aren't particularly illuminating for knowing where it comes from.
1789 G. Parker Life's Painter xiv. 130 The cull with whom she snooz'd.
Using Eighteenth Century Collections Online (university subscription), I found several early entries.
The 1753 edition of The discoveries of John Poulter, alias Baxter by J. Poulter include a brief cant lexicon that includes the following:
The Cull is at Snoos ; the Man is at Sleep
In The festival of Momus, a collection of comic songs, including the modern and a variety of originals (1780), this bit appears (p. 69):
I snooze at the hummums (a Turkish bathhouse) 'till twelve...
Snooze also appears in some other song books. Here's The Town and country songster's companion (1780-1800), p. 37:
All higley pigley, pigs in the straw, / We snooze without thinking of harm, 'Till a signal that goes, / We jump into our cloaths, / And as fine a confusion as ever you saw / Is the midnight false alarm.
By 1795, Humphry Tristram Potter includes snooze in his A new dictionary of all the cant and flash languages, both ancient and modern. While Potter's lexicography involves a lot of plagiarism and unreliability (as Julie Coleman claims in A History of Cant and Slang Dictionaries Volume II), given the evidence above, snooze appears to have persisted in cant and in other lower registers like popular lyric:
SNOOZE, to sleep
It may be related to other sn- nose words (snout, snore) or sleep-related words (doze, drouse, snug), but that's guesswork.