How does one pronounce "nihilism"?
Solution 1:
I'm afraid I vary my pronunciation of "nihilist / nihilism" to suit the sentence and my audience. "Nee-(h)il-ist", with a very soft "h", feels most correct to me, but for people who haven't studied philosophy (or who forgot it as soon as they left college), "ny-il-ist" is easily recognized from "annihilation" (I have never heard anyone pronounce that word as "an-nee-(h)il-ation").
My favorite sighting (hearing?) of "nee-(h)il-ist" is in the (NSFW) song 88 Lines About 44 Women by The Nails; at about 4:12 he mentions #39:
Terri didn't give a s**t
She was just a nihilist
Solution 2:
Nile-ism. (at least in North America)
Solution 3:
Something to remember is
it's just like annihilate,
which has a pretty agreed pronunciation.
As a child, I thought nihilism was pronounced "nil-ism" (the "nil" makes sense in terms of the meaning of the word) as I had only read it and never heard it!
Solution 4:
The OED says "nigh-ill-ism", but Latin would have been "knee-hill-ism", so I suppose you can just choose the one you want, and back it up with the one that supports it.
On the extremely few occasions when I've said it, I've gone with "nigh~~".
Solution 5:
Either nih-hil-ism (patterned after the latin from which it derives), or naye-il-ism or naye-hil-ism.