Disclaimer: I don't know a lot about modules except that they're a ring acting on an abelian group such that multiplication in our ring becomes composition and „both additions commute“. Also, I hope this question is not too „vague“ for MSE.
Idea
Since we often look at either substructures or homomorphic images, I was wondering if a (surjective) ring homomorphism $R\stackrel{h}{\to} H = R/J$ with $J:= \ker h$ induces a related module structure ($R$ commutative with identity). Then perhaps we could use the lattice of ideals of $R$ to gather information about our module.
So consider a Left $R$-module $M$ with multiplication represented by the ring action $$\lambda\colon R\times M\to M, \quad (r,m)\mapsto \lambda_rm.$$ Let now $(p):=J \unlhd R$ be a principal ideal, which is necessary in order for $$\lambda_JM:=\{\lambda_jm\mid j\in J, m\in M\}$$ to be a submodule which we can quotient out: We're gonna turn $M/\lambda_JM$ into an $H$-Module.
Define $$ \lambda_H\colon H\times (M/\lambda_JM)\to M/\lambda_JM,\\ \lambda_{[r]} [m] := [\lambda_r m],$$ where $[r]=r+J\in H=R/J$ and $[m]=[n]\Leftrightarrow m-n\in \lambda_JM$ denote the equivalence classes discussed above.
Lemma $\lambda_H$ is a ring action.
Proof. Well-definedness: let $r\in R,\; i,j\in J,\; m,n\in M$. We have $$ \lambda_{[r+i]}[m+\lambda_jn] = [\lambda_{r+i}(m+\lambda_jn)] = [\lambda_rm +\underbrace{\lambda_im}_{\in \lambda_JM} +\underbrace{\lambda_{rj}n}_{\in \lambda_JM} +\underbrace{\lambda_{ij}n}_{\in \lambda_JM}] = [\lambda_rm] = \lambda_{[r]}[m]. $$ Ring action: directly follows from $\lambda$ being a ring action by easy calculation.
Question
To give a close-ended question: Is there a name for this construction? If yes, where is it used/introduced?
And, as a more open-ended follow-up: What properties of $M$ are preserved under this construction (finitely generated, free, torsion-free, …)? What can we say about the module structure of $M$ knowing the ideal structure of $R$?
They don't really have a name, but modules $M/IM$ occurs very often in commutative algebra and algebraic geometry, and is a special case of change of rings $M\otimes_R S$ for some ring homomorphism $R\to S$. You can talk about all sorts of property of $M$ that is invariant under base change, e.g. projective, faithful flatness, finitely-generated, etc.