What is the origin of the phrase "not to mention ..."

The rhetorical, as opposed to etymological, origin is the device known as paralpsis, paraleipsis, paralepsis, (also praeteritio) meaning pretended omission for rhetorical effect, because in saying we won't mention X, of course we just did.

Edit (by FumbleFingers): A later question on the same topic gives the relevant word as apophasis. It's a little hard to see from this graph, but what it shows is that until recently, the combined total written instances for various spellings paralpsis, paraleipsis, paralepsis dwarfed those of apophasis. But the graph from 1970 on shows apophasis is now overwhelmingly more common. I don't think this means the correct term for this rhetorical device has just changed - it's just that the theological use of apophatic has led to it being more commonly known, and used "metaphorically".

Although it's still not clear what it means to ask for the etymology of the phrase, I can give some early examples of its usage. (Found through looking on Google Books for the 1600s. If you try this yourself, be sure to check the dates and not trust Google.)

Here's William Somner in 1640 (BTW, 'ſ' is the long s):

… their workes like themſelves ſpeaking Engliſh. The learned, I know, can further ſatisfie themſelves in the point from Bertherius in his Pithanon, and Morinus of late, in Eccleſiaſtica Exercitationes, not to mention divers others.

So here it's very much the literal meaning: there are "divers others", but they have not been mentioned. You can find many other usages of this type, where "not to mention" is not a rhetorical device of pretended omission, but used merely to point out that certain things have not been explicitly mentioned by name.

From here it's a small step (since the form is superficially very similar) to the many usages where the others are mentioned by name (a random example: this from 1698). (I'm not claiming this is how it happened, though.)

The rhetorical usage is also seen in a 1682 work by Robert Boyle, part of his disputes with Thomas Hobbes:

For, not to mention that it is ſtill by many learned men doubted whether the Terreſtrial Globe it ſelf have it; nor to examine whether or no he aſſigns a good Natural cauſe of it; it is not always true that… (p. 28)
[…]
For, not to mention that the Argumentation is invalid, unleſs by Fluidum he mean Omne fluidum, I reply … (p. 83)