I have the following situation: Let $C$ be a category and $X\in {ob}(C)$
Let $C_X$ denote the category whose objects are monic arrows into $X$ and whose morphisms are commuting triangles in $C$ (where the underlying morphism between the domains needs not be monic)
There's the following property of $X$:
The limit of any diagram $F:\mathbb{\omega}^{op}\rightarrow C_X$ exists, and is given by $F(n)$ for some $n\in\omega$ and all higher morphisms are identities
This generalizes classical Noetherianess of objects, that asserts the same, only in the category whose morphisms are monic triangles.
Is there a name for this generalization?
Unless I'm misunderstanding something, your proposed generalization is not actually a generalization; your property is equivalent to Artinianness (and not Noetherianness : you would need to replace the limit of $F: \mathbb{\omega}^{op} \rightarrow C_X$ by the colimit of $F:\mathbb{\omega}\rightarrow C_X$ for that).
Indeed, if I understand correctly the objects of your $C_X$ are monomorphisms, and an arrow between two objects $m:M\to X$ and $n:N\to X$ is an arrow $f:M\to N$ of $C$ such that $n\circ f=m$. You say that $f$ need not be monic, but in fact it always is in that case : if you have two arrows $u,v$ for which $f\circ u=f\circ v$, then $$m\circ u=n\circ f\circ u=n\circ f\circ v=m\circ v,$$ and thus $u=v$ since $m$ is a monomorphism.
So actually your commuting triangles will always be monic triangles, so your property is in fact the same as Artinianness.