If The Singularity is as close as some people say, you don't have much time left to think up and spread a new term, and the current term is pretty well established. As an Esperantist, I appreciate the difficulty of tilting at linguistic windmills. ;)

In this sentence, I disagree with the first assertion, and find the second irrelevant:
"So the term is wrong for those who understand the mathematical meaning and meaningless to those who don't."

In mathematics, "singularity" has connotations of not merely "undefined" but "goes to infinity":
http://en.wikipedia.org/wiki/Singularity
with the classic singularity being division by x as x goes to zero.

That it is meaningless to people who don't understand 1/x is not very important, as 1) they probably don't care about the subject, 2) they can learn the meaning, and 3) they just need a pronouncable label that hasn't yet been taken.

I rather like the connotations of the capitalized word preceded by the definite article: not a singularity, but The Singularity.

But denying the existence of an alternate solution is easy, boring, and non-creative, so let me take a stab at it. How about The Emergence? That suggests something about the nature of what is about to happen, though by the rules we are not supposed to be able to predict it. However, perhaps there are some underlying rules that govern all these phase changes. I like Robert Wright's proposal in Nonzero:
http://www.nonzero.org