= Solution
Whenever the graded pieces $N_i$ are finitely generated over $S_0$, define the <Poincare series of a graded module> by
$$
P_{N,\lambda}(t)=\sum_{i\ge0}\lambda(N_i)t^i.
$$
Additivity means $\lambda(B)=\lambda(A)+\lambda(C)$ for every <short exact sequence> $0\to A\to B\to C\to0$ in the indicated category. Dimension over a <field> is a basic example; length is another when the <modules> have finite length.
The <Hilbert-Serre theorem> needs finite-generation hypotheses: take $S_0$ commutative <Noetherian> and $S$ generated over it by homogeneous $x_1,\ldots,x_r$ of positive degrees $d_1,\ldots,d_r$. For finitely generated graded $N$, every $N_i$ is then finite over $S_0$, and
$$
\boxed{P_{N,\lambda}(t)=\frac{Q(t)}{\prod_{j=1}^r(1-t^{d_j})},\qquad Q(t)\in\mathbb Z[t]}.
$$
If negative grading indices are allowed for the <module>, the numerator is a <Laurent polynomial> instead. In standard degree one, the coefficients are eventually <polynomial> functions of the index.
To prove this, regard $N$ as finite over the surjective <polynomial> presentation $S_0[X_1,\ldots,X_r]\to S$. Induct on $r$. At $r=0$ only finitely many homogeneous generator degrees occur, so the series is a <polynomial>. For the last variable $x$ of degree $d$, put $K=(0:_Nx)$ and $Q=N/xN$. The <polynomial ring> is <Noetherian> by the <Hilbert basis theorem>, hence $K$ is finite; both $K$ and $Q$ are finite <modules> over the <ring> with that variable removed. The degree-$i$ exact sequence is
$$
0\longrightarrow K_{i-d}\longrightarrow N_{i-d}\xrightarrow{x}N_i\longrightarrow Q_i\longrightarrow0.
$$
Additivity, followed by summing over $i$, gives
$$
(1-t^d)P_{N,\lambda}=P_{Q,\lambda}-t^dP_{K,\lambda}.
$$
The induction hypothesis supplies the remaining denominator factors. This proves <Hilbert-Serre theorem for an additive coefficient function>. If all $d_i=1$, expanding $(1-t)^{-r}$ proves eventual <polynomial> coefficients directly.
An arbitrary positively <graded ring> in the introductory wording is not enough by itself. For example $S=k[X_1,X_2,\ldots]$ with every variable of degree one and $N=S$ is a cyclic <graded module>, but $N_1$ is not finite-dimensional over $k$, so the dimension-valued series is not even defined. The theorem above states the standard hypotheses that make the requested definition and rationality valid.
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