Here is a idea concerning the classification of finite rings (commutative, unital). Related question: Classification of finite commutative rings.

Every finite ring is a direct product of finite algebras over $\mathbb{Z}/p^n$ for some prime power $p^n$ (Chinese Remainder Theorem). Now fix such a prime power. If $R,S$ are finite algebras over $\mathbb{Z}/p^n$, then $R \otimes_{\mathbb{Z}/p^n} S$ and $R \times S$ are also finite algebras over $\mathbb{Z}/p^n$. If $\mathcal{A}$ is the set of isomorphism classes of finite algebras over $\mathbb{Z}/p^n$, then $\mathcal{A}$ becomes a commutative semiring with addition $\times$, zero element $0$, multiplication $\otimes$ and identity $\mathbb{Z}/p^n$. What is known about the associated ring $\overline{\mathcal{A}}$? Note that the map $\mathcal{A} \to \overline{\mathcal{A}}$ is not injective (I don't think that $\mathcal{A}$ is cancellative), but perhaps this loss of information makes it possible to give a desciription of this ring.

What about varying $n$? For every $k \leq n$ we may regard $\mathcal{A}_k$ as an ideal of $\mathcal{A}_n$, and $R \mapsto R/p^k$ is a projection $\mathcal{A}_n \to \mathcal{A}_k$. Does this help to describe $\mathcal{A}_n$ or $\overline{\mathcal{A}_n}$ recursively?