I really think names without a fairly obvious way of saying them out-loud should just be avoided/deprecated. I can't pronounce this as "ring" (because then you'd think I was referring to a ring, not a rng). Do I just spell it out? R-N-G? I'd rather just call it a non-unital ring. That involves more letters but is much more descriptive (to a mathematician, anyway).
"If we may speak frankly, the rig and rng nomenclatures are abominations. Nevertheless, you may see them sometimes, but we will speak of them no more."
:)
But 'nearring' doesn't bother you? If I'm at a talk on rngs where the distinction from rings is important, then I'll know that's the kind of talk that I'm attending, and hear accordingly; or else the speaker will make a huge deal of it (and then probably not use the word).
If it's so important to have a "rng" instead of a "ring" then using its name in some example of its usage should probably be better.