I would just like to verify I have the right idea here. The problem is to classify up to isomorphism all modules having precisely 8 elements over $\mathbb{Z}_2[x]$.
Since $\mathbb{Z}_2[x]$ is a PID, we can use the Fundamental Theorem for Finitely Generated Modules over PID's, and the modules are isomorphic if and only if they have the same lists of invariant factors.
First, the case where there is a single invariant factor. Then it must be a degree 3 polynomial, so $a(x)=x^3+Ax^2+Bx+C$ where each of $A, B, C$ equals either $0$ or $1$. This gives $8$ non-isomorphic modules.
Now, suppose we have 2 invariant factors. Then the first invariant factor must be of degree 1, and the second of degree 2. Quickly we find that the possible lists are : $\{ x, x(x+1)\}, \{ x, x^2\}, \{(x+1), x(x+1)\}, \{x+1, (x+1)^2\}$ giving us another $4$ non-isomorphic modules.
Finally, if we have 3 invariant factors, they each must be of degree 1, hence the two possible lists are: $\{x, x, x\}$ and $\{x+1, x+1, x+1\}$ giving us another $2$ non-isomorphic modules.
Hence, we have a total of $14$ non-isomomorphic modules with eight elements.
Did I miss anything? MAny thanks!