Any field $R$ in the stable homotopy category (ie. a homotopy associative ring spectrum such that $\pi_{*}R$ is a field) admits a structure of a $K(n)$-module for some $0 \leq n \leq \infty$, in particular as a spectrum it is a direct sum of $K(n)$, this follows from people.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf and the nilpotence theorem. Thus, $K(n)$ are "minimal" fields in this sense and this property identifies them uniquely.

Thus, one way to make sense of the notation $K(\infty) = H \mathbb{F}_{p}$ is to observe that the Eilenberg-MacLane spectrum is the "a minimal $p$-local field whose associated formal group is of height $\infty$" and the Morava $K$-theories are "minimal p-local fields of finite height".