Why are Noetherian Rings Special?

In short, "Noetherian-ness" is a property which generalizes "PID-ness." As Keith Conrad so nicely puts it,

The property of all ideals being singly generated is often not preserved under common ring-theoretic constructions (e.g. $\mathbb{Z}$ is a PID but $\mathbb{Z}[x]$ is not), but the property of all ideals being finitely generated does remain valid under many constructions of new rings from old rings. For example... every quadratic ring $\mathbb{Z}[\sqrt{d}]$ is Noetherian, even though many of these rings are not PIDs." (italics added)

So you see? We like rings with finitely generated ideals because it keeps the math (relatively) nice. For example, you could ask, "Given a Noetherian ring $R$, can I build a new ring such that it, too, is Noetherian?" Yep. You can construct the polynomial ring $R[x]$ and it will be Noetherian whenever $R$ is. For more on the Noetherian property, see here.