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I am reading a cryptography paper, and the authors introduce a function $f(x) = x^{-\omega(1)}$ and call it a negligible function in $x$. What is the possible meaning of this?

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This is a relative of big-O notation. We say that $f(x) = \omega(g(x))$ if $f(x) \ge k g(x)$ for all positive constants $k$, at least asymptotically.

So the notation should really be read $- \log f(x) = \omega(\log x)$, or $- \log f(x) \ge k \log x$ for every positive constant $k$ asymptotically. In other words, $f(x) \le x^{-k}$ for every positive constant $k$ asymptotically, hence $f$ vanishes faster than polynomially as $x \to \infty$.

Keep in mind that this does not define a function in the usual sense; rather, it specifies the asymptotic behavior of that function in a way that is good enough for whatever the paper is proving.

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