I was trying to prove this statement as true. Actually I found an identical question here: "Is every Artinian module over an Artinian ring finitely generated?"
However, in the proof of the link above, a key step is
Claim:"If $M$ is not finitely generated one can assume that every proper submodule of $M$ is finitely generated. (In order to see this take the partial ordered set of submodules of $M$ which are not finitely generated and choose a minimal element.) "
I suppose that the statement above assumes we can have a minimal generating set for each submodule. However, I cannot prove that.
Or in another way, the claim above is used to prove that $Ann(M)$ is a prime ideal in $R$, so, is their any way to prove $Ann(M)$ is a prime ideal in $R$?
For some information which might be useful, we have: a commutative Artinian ring is Noetherian.(http://math.stanford.edu/~conrad/210BPage/handouts/Artinian)
Modules whose proper submodules are finitely generated is called almost finitely generated http://www.sciencedirect.com/science/article/pii/0021869383900753 In this article it seems that the claim is not true for a general ring, however, I don't know how to use $R$ being Artinian in the proof.
I do not have enough reputation to comment on the original question so have to open a new one.