What is a finitely generated sheaf? I do exercise asking to show on Noetherian space $X$, any subsheaf $\mathscr{R} \subseteq \Bbb{Z}_U$ is finitely generated. $\Bbb{Z}_U$ is sheaf $i_{!}(\Bbb{Z}|_U)$ for $U \subseteq X$ open.  What is the meaning of a finitely generated sheaf? 
Added: Hartshorne does not even define finitely-generated sheaf. What does this even mean?
 A: The statement of the exercise you quote is incorrect : $\mathbb Z_U$ itself is  not  finitely generated except in trivial cases.    
For example if $X=\mathbb A^1_k$ and $U=X\setminus \{O\}$ (with $O$ the origin), the stalk at $O$ of  $\mathbb Z_U$ is zero: $\mathbb (Z_U)_O=0$.
If  $\mathbb Z_U$ were finitely generated its stalks would be zero in a neighbourhood of $O$.
But in reality all the stalks   $ (\mathbb Z_U)_x=\mathbb Z$ for $O \neq x\in X$ and thus it is false that  $\mathbb Z_U$ is a finitely generated sheaf of abelian groups.
Edit
Following the friendly discussion in the comments, let me add that the last line of  EGA I page 45 states that the support of a finitely generated sheaf is closed. Since the support of $\mathbb Z_U$ is $U$, this confirms that $\mathbb Z_U$ can only be finitely generated in the rare case that $U$ is both open and closed.
A: If $(X,\mathcal{O}_X)$ is a ringed space, then there is a well-known notion of $\mathcal{O}_X$-modules of finite type (Stacks Project, Tag 01B4). Any topological space $X$ can be endowed with the constant sheaf $\mathcal{O}_X:=\mathbb{Z}_X$ so that $\mathcal{O}_X$-modules coincide with sheaves of abelian groups on $X$.
A: Based on Hartshorne's hints, I believe what he means is (a variation on) the category-theoretic notion of "finitely presentable":

A finitely presentable object in a category $\mathcal{C}$ is an object $A$ such that the functor $\mathrm{Hom}(A, -) : \mathcal{C} \to \mathbf{Set}$ preserves directed/filtered colimits.

Let us take $\mathcal{C}$ to be the category of (abelian) sheaves on a noetherian topological space $X$. Then $\mathrm{Hom}(\mathbb{Z}_U, -)$ is isomorphic (as a functor) to $\Gamma (U, -)$. But the property of being noetherian is hereditary, so we may use [Chapter II, Exercise 1.11] to deduce that $\Gamma (U, -)$ preserves directed colimits. Thus $\mathbb{Z}_U$ is indeed finitely presentable.
That said, Hartshorne speaks of subsheaves of $\mathbb{Z}_U$ as well. It is not at all clear to me that these are finitely presentable. Nor is it clear to me that a sheaf with the extension property with respect to all subsheaves $\mathscr{R} \subseteq \mathbb{Z}_U$ and all open subsets $U \subseteq X$ is necessarily injective. Certainly what is true is this: if $\mathcal{I}$ is a collection of monomorphisms such that every monomorphism in $\mathcal{C}$ can be obtained as a retract of some $\mathcal{I}$-cell complex (= a transfinite composition of pushouts of $\mathcal{I}$), then an injective object in $\mathcal{C}$ is the same thing as an object with the extension property with respect to $\mathcal{I}$; and if $\mathcal{I}$ consists of morphisms whose domain and codomain are finitely presentable, then the class of $\mathcal{I}$-injective objects is closed under directed/filtered colimits.
