We call a ring with a unique maximal (left/right) ideal $\textit{local}$. Do we have common terminology for modules that have a unique maximal submodule? It would seem to make sense to call them local modules, but this (as far as I am aware) does not seem to be in common use.
Is there a particular reason why?
In this solution I cited three works defining “local module.”
I think it is a perfectly reasonable thing to define one way or the other, it’s just a question of how useful it is for what you are writing. Unlike for rings, there is some complication because sometimes sub modules need not be contained in maximal submodule a, and therefore a unique maximal submodule might not contain all other proper submodules.