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The movie "The thing" was "Нéчто" in Russian. "Что-то" wouldn't work just as good.

No, you should not.
Нéчего is the genitive of a word that doesn't have the nominative at all.
Héчто, which looks like the nominative of нечего, is the nominative of a different word, which doesn't...

Hmm. I can't imagine how нEчто could possibly mean nothing. It's ничтО which is nothing.

I never heard of such a thing, but maybe I'm just lucky. :)
And I don't even live in Russia.

I agree.
Another option, not mentioned yet:
существует нечто, называемое