Why are noetherian rings such natural objects in algebraic geometry?
The best answer I've ever been able to come up with is that the class of noetherian rings contains the classical number rings $\mathbf{Z}$ and $\mathbf{R}$ and is closed under the formation of polynomial rings, localization, completion, and quotients. So it contains many of the rings you will come across in ordinary situations (whatever that means). It also has the advantage that the definition is tractable enough that if someone hands you an explicit ring, it's not out of the question to try to work out from scratch whether it's noetherian. If you're the kind of person who likes abstract fields, then they're also included.
On the other hand, I don't think of it as a truly fundamental concept, like say finite presentation. But there is no denying its convenience. If you need to avoid some infinitary phenomena but you still want a broad class of rings, it's often hard to beat noetherianness. It's also quite good in situations where you're too lazy to work out exactly what finiteness conditions you care about.
I am not entirely sure what you have in mind when you contrast being noetherian to being graded. These belong to different aspects of being a ring. It's kind of like saying "He came in a hurry and [in] a winter coat".
Also, graded algebras are actually natural objects in algebraic geometry. That's where projective schemes/varieties come from.
Anyway, let me say something possibly useful, too: I believe that the truly important notion is being finitely presented. See the original definition of coherence. If you accept that, then one can say that the importance of being noetherian is that being finitely presented follows from being finitely generated and it is inherited by both quotients and subobjects.
I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus
- The regular (non-singular) locus can fail to be open.
- Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the different lengths).
- Normalization need not be a module-finite extension.
Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies and some others (and avoid these after some standard operations).
The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as are complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.