Despite knowing how the phrase "not to mention ..." is often used, it still grates on me to use it because I am in the act of "mentioning" even as I use it. I found it helpful to read the origins of the phrase here, but I would really like to use an alternative word or phrase that means the same thing. Any suggestions?
I'd consider not to mention as a type of scalar focus marker. It cannot be replaced by any of the suggested wordings in the other answers without stripping it of its special focus-marking properties.
X not to mention Y can be paraphrased as:
"X should be considered noteworthy; and if you don't think so, there is the related fact Y which is possibly even more noteworthy than X is."
"X supports my argument. If X does not support my argument, then there is Y which supports my argument a fortiori.
Don't leave it out of your repertoire for such silly reasons as its literal reading.
How about using "last but not least" or simply "plus?'
They own two houses, a yacht and, last but not least, a jet.
Also, consider the expression "let alone" for negative statements.
We have no room for another houseguest, let alone [=even less for] an entire family.
There may not be an exactly equivalent phrase, but there may be some that work about as well in certain contexts. For instance, "more importantly" may work when the statement it introduces carries extra weight specifically because of its importance.
Personally, I sometimes use "moreover" in this way; although its standard meaning doesn't actually indicate emphasis on the item it introduces, it's a rare enough word that it provides emphasis simply by being conspicuous, and I tend to think that the presence of the word "more" at the beginning of the word "moreover" probably provides a slight emphasis in the reader's mind even if that's not part of the standard definition.
(I also tend to use "furthermore" similarly, although in my mind it provides somewhat less emphasis.)