As Morten Brun said, the big Witt vector functor is the right adjoint of the forgetful functor from the category of lambda-rings to the category of rings (commutative). But this answer is not completely satisfying in that Witt vectors usually come up in number theory, in contexts that have little direct connection to K-theory. It's also not clear what the analogue of that statement for the p$p$-typical Witt vectors is. (p$p$ is a prime here. The p$p$-typical Witt vectors are the usual "non-big" Witt vectors as defined by Witt and which come up in the theory of local fields, for instance.)
To me, the most satisfying answer to this question is that Witt W(A)$W(A)$ gives the universal way of equipping your ring A$A$ with lifts of Frobenius maps. For simplicity, let's look at the p$p$-typical Witt vectors. Then W(A)$W(A)$ has a ring endomorphism F$F$ which is congruent to the p$p$-th power map modulo the ideal pW(A)$pW(A)$. In other words, W(A)$W(A)$ has a lift of the Frobenius endomorphism of W(A)/pW(A)$W(A)/pW(A)$. There is also a ring map W(A)-->A$W(A) \rightarrow A$ given by projection on the the first component.
Now, in what sense is W(A)$W(A)$ universal? Suppose B$B$ is another ring equipped with a ring map B-->A$B \rightarrow A$ and an endomorphism F:B-->B$F:B \rightarrow B$ lifting the Frobenius endomorphism of B/pB$B/pB$. Then, assuming B$B$ is p$p$-torsion freetorsionfree, there exists a unique ring map B-->W(A)$B \rightarrow W(A)$ commuting with the two maps F$F$ and the two maps to A$A$. (This is a theorem called "Cartier's Dieudonne-Dwork lemma" in classical exposition of the Witt vectors, but is essentially true by definition in some more recent ones.) Thus, ignoring the issue of p$p$-torsion, W(A)$W(A)$ is the universal ring mapping to A$A$ with a Frobenius lift.
How do we deal with torsion? First, if A$A$ itself is p$p$-torsion free, then so is W(A) -$W(A)$ - it is actually a subring of an infinite product of copies of A$A$. So then W$W$ is the right adjoint of the forgetful functor from the category of p-torsion$p$-freetorsionfree rings equipped with a Frobenius lift to the category of p-torsion$p$-freetorsionfree rings. Now it will one day be clear that the most important uses of W(A)$W(A)$ are when A$A$ is torsion free, but certainly the most important existing applications are when A$A$ is an F_p$\mathbb{F}_p$-algebra, where everything is p$p$-torsion. So it would be nice to have a universal property that works whether there is p$p$-torsion or not.
Probably the most straightforward way to do this is to use a better definition of "Frobenius lift". If F$F$ is a Frobenius lift on B$B$ as above and B$B$ is p$p$-torsion free, then d(x)=(F(x)-x^p)/p$d(x)=(F(x)-x^p)/p$ is a well-defined operator on B$B$. The condition that F$F$ be a ring endomorphism can be of course be expressed in terms of slightly complicated identities on d$d$. The key point, then, is that by the magical properties of binomial coefficients modulo p$p$, these identities have integral coefficients -- there are no p's$p$'s in the denominators! Then we can define a d$d$-ring structure on any ring to be an operator d$d$ satisfying these conditions. Then you can show by reduction to the p$p$-torsion-free case that W$W$ is the right adjoint of the forgetful functor from the category of d$d$-rings to the category of rings. The point of all this is to eliminate the existential quantifier hidden in the word "lift" by specifying a y$y$ such that F(x)-x^p$F(x)-x^p$ is p$p$ times y$y$, rather than just saying some such element y$y$ exists.
