In the chapter I.3.3 of Eisenbud & Harris "The Geometry of Schemes" they give a definition of "prime ideal sheaf":
Let $\mathcal{O}_S$ be the structure sheaf of a scheme $S$, and let $\mathcal{F}$ be a quasicoherent sheaf of $\mathcal{O}_S$-algebras. Then a prime ideal sheaf $\mathcal{I} \subset \mathcal{F}$ is a quasicoherent sheaf of ideals of $\mathcal{F}$, such that on eash open affine $U \subset S$ the ideal $\mathcal{I}_U$ is prime (or unit).
After that they claim that prime ideal sheaves of $\mathcal{O}_S$ itself correspond to points of $S$. And here probably I miss some point, since I don't see why the following is not a counterexample.
Consider the following non-separated scheme, presented earlier in the same book. Let $S$ be a scheme that is obtained by gluing two copies of the affine line $S_1 \simeq S_2 \simeq \mathrm{Spec} \, k[X]$ along the complement of $0$ (I mean the ideal $(X)$, of course) by identity morphism, and let $0_1$ and $0_2$ be the "two zeroes" of $S$. Then I think that the sheaf of ideals corresponding to the reduced two-point subscheme supported on $\{0_1, 0_2\}$ satisfies the definition of prime ideal sheaf, since no affine open subset can contain both points simultaneously. However, I don't think this sheaf of ideals deserves to be called prime, and it clearly violates their claim.
So where am I wrong? Or if it is really a counterexample, what is the right definition then? Is it enough to add that $\mathcal{F}/\mathcal{I}$ should have irreducible support?
Edit: It turned out that in the (more recent) paper edition they only claim that for affine schemes.
I think your counterexample is correct (and this also means that Eisenbud's Spec construction breaks down; by the way there is a global Spec construction which even works for locally ringed spaces, see here). The correct definition of a prime ideal sheaf is that the corresponding closed subscheme is integral. Equivalently, it should be irreducible and reduced. Whereas reducedness can be checked locally, this is not the case for irreduciblity. So here is a more down-to-earth description: $I \subseteq \mathcal{O}_S$ is prime if $I \neq \mathcal{O}_S$, for every open affine $U \subseteq S$ the ideal $\Gamma(U,I) \subseteq \Gamma(U,\mathcal{O}_S)$ is prime or $(1)$, and for every inclusion $V \subseteq U$ of open affines the map $\Gamma(U,\mathcal{O}_S) / \Gamma(U,I) \to \Gamma(V,\mathcal{O}_S) / \Gamma(V,I)$ is injective.
PS: You may contact Eisenbud. He collects errata for his book.