Algebraic K-theory originated in classical materials that connected class groups, unit groups and determinants, Brauer groups, and related things for rings of integers, fields, etc, and includes a lot of local-to-global principles. But that's the original motivation and not the way the work in the field is currently going - from your question it seems like you're asking about a motivation for "higher" algebraic K-theory.
From the perspective of homotopy theory, algebraic K-theory has a certain universality. A category with a symmetric monoidal structure has a classifying space, or nerve, that precisely inherits a "coherent" multiplication (an E_oo-space structure, to be exact), and such an object has a naturally associated group completion. This is the K-theory object of the category, and K-theory is in some sense the universal functor that takes a category with a symmetric monoidal structure and turns it into an additive structure. The K-theory of the category of finite sets captures stable homotopy groups of spheres. The K-theory of the category of vector spaces (with appropriately topologized spaces of endomorphisms) captures complex or real topological K-theory. The K-theory of certain categories associated to manifolds yields very sensitive information about differentiable structures.
One perspective on rings is that you should study them via their module categories, and algebraic K-theory is a universal thing that does this. The Q-construction and Waldhausen's S.-construction are souped up to include extra structure like universally turning a family of maps into equivalences, or universally splitting certain notions of exact sequence. But these are extra.
It's also applicable to dg-rings or structured ring spectra, and is one of the few ways we have to extract arithmetic data out of some of those.
And yes, it's very hard to compute, in some sense because it is universal. But it generalizes a lot of the phenomena that were useful in extracting arithmetic information from rings in the lower algebraic K-groups and so I think it's generally accepted as the "right" generalization.
This is all vague stuff but I hope I can at least make you feel that some of us study it not just because "it's there".
The Eilenberg-Zilber theorem says that for singular homology there is a natural chain homotopy equivalence:
$$S_*(X)\otimes S_*(Y) \cong S_*(X\times Y)$$
The map in the reverse direction is the Alexander-Whitney map. Therefore we obtain a map
$$S_*(X)\rightarrow S_*(X\times X) \rightarrow S_*(X)\otimes S_*(X)$$
which makes $S_*(X)$ into a coalgebra.
My source (Selick's Introduction to Homotopy Theory) then states that this gives $H_*(X)$ the structure of a coalgebra. However, I think that the Kunneth formula goes the wrong way. The Kunneth formula says that there is a short exact sequence of abelian groups:
$$0\rightarrow H_*(C)\otimes H_*(D) \rightarrow H_*(C \otimes D) \rightarrow \operatorname{Tor}(H_*(C), H_*(D)) \rightarrow 0$$
(the astute will complain about a lack of coefficients. Add them in if that bothers you)
This is split, but not naturally, and when it is split it may not be split as modules over the coefficient ring. To make $H_*(X)$ into a coalgebra we need that splitting map. That requires $H_*(X)$ to be flat (in which case, I believe, it's an isomorphism).
That's quite a strong condition. In particular, it implies that cohomology is dual to homology.
Of course, if one works over a field then everything's fine, but then integral homology is so much more interesting than homology over a field.
In the situation for cohomology, only some of the directions are reversed, which means that the natural map is still from the tensor product of the cohomology groups to the cohomology of the product. Since the diagonal map now gets flipped, this is enough to define the ring structure on $H^*(X)$.
There are deeper reasons, though. Cohomology is a representable functor, and its representing object is a ring object (okay, graded ring object) in the homotopy category. That's the real reason why $H^*(X)$ is a ring (the Kunneth formula has nothing to do with defining this ring structure, by the way). It also means that cohomology operations (aka natural transformations) are, by the Yoneda lemma, much more accessible than the corresponding homology operations (I don't know of any detailed study of such).
Rings and algebras, being varieties of algebras (in the sense of universal or general algebra) are generally much easier to study than coalgebras. Whether this is more because we have a greater history and more experience, or whether they are inherently simpler is something I shall leave for another to answer. Certainly, I feel that I have a better idea of what a ring looks like than a coalgebra. One thing that makes life easier is that often spectral sequences are spectral sequences of rings, which makes them simpler to deal with - the more structure, the less room there is for things to get out of hand.
Added Later: One interesting thing about the coalgebra structure - when it exists - is that it is genuinely a coalgebra. There's no funny completions of the tensor product required. The comultiplication of a homology element is always a finite sum.
Two particularly good papers that are worth reading are the ones by Boardman, and Boardman, Johnson, and Wilson in the Handbook of Algebraic Topology. Although the focus is on operations of cohomology theories, the build-up is quite detailed and there's a lot about general properties of homology and cohomology theories there.
Added Even Later: One place where the coalgebra structure has been extremely successfully exploited is in the theory of cohomology cooperations. For a reasonable cohomology theory, the cooperations (which are homology groups of the representing spaces) are Hopf rings, which are algebra objects in the category of coalgebras.
Best Answer
A student of mine asking for a motivation unmotivated by applications? Haven't I taught you anything, Mike? (Joking of course.) However, perhaps one way to avoid talking about future applications is to reflect on implicit past applications and the explicitly stated original motivations. This is not to take away from your answer, Craig, you know I agree completely, but I'll give an answer orthogonal to yours.
We know from long past history that focusing on some generalized theory can drastically simplify life. Perhaps the first and still one of the most persuasive examples is comparison of the solutions of the Hopf invariant one problem by $K$-theory and by mod $2$ cohomology. This, together with the knowledge that $K$-theory captures so much of concrete importance (index theorem, etc) certainly argues for focusing on $K$-theory. The immense calculational power of periodicity phenomena also argues for a setting that throws away anything that is not periodic. The search for understanding periodicity phenomena in the stable homotopy groups of spheres goes back more than half a century, and the expectation that localizations at homology theories would be relevant goes back almost as far.
In fact, Frank Adams explicitly conjectured (Conjecture 4.6) in his 1973 Chicago lecture notes "Localization and completion" that localizations with respect to generalized homology theories exist. Pete Bousfield attended those lectures and constructed the conjectured localizations the following year. (I attended too and I remember Pete tapping me on the shoulder and asking "How does he know it is a set?"). In fact, Adams' notes fail to give a proof only because of set theoretic questions. Zig Fiedorowicz has uploaded a version of those notes on the arXiv (http://arxiv.org/abs/1012.5020) and he has added an epilogue showing that an easy modification of the argument Adams originally had in mind proves the conjecture, and "has thus taken the liberty of upgrading" the statement, so that it appears as Theorem 4.6 in the ArXiv version of the notes. To quote from Zig's foreward ``Thus it can now be seen in retrospect that Adams amazingly succeeded in his project of "constructing localizations and completions without doing a shred of work".
Frank's lectures were in a sense all about his motivation for wanting such localizations. So, philosophically, I might argue that motivations for mathematical developments might best be sought in their historical context, rather than abstractly, even for a category theorist. But a category theorist might like what Frank says right at the start (p. 10) "I want to study functors in homotopy theory with the same formal properties as localization, so I'd better say what those properties are". And his list of axioms is quite satisfactorily categorical.