1. Here is what Tate says in his account of the General Reciprocity Law in the AMS volume on Hilbert's problems :
With this work of Takagi the theory of
abelian extensions --- "class field
theory" --- seemed in some sense
complete, yet there was still no
general reciprocity law. It remained
for Artin to crown the edifice with
such a theorem. He conjectured in
1923 and proved in 1927 that there is
a natural isomorphism $$
> C_K/N_{L|K}C_L\buildrel\sim\over\to\operatorname{Gal}(L|K)
> $$ which is characterised by the fact
that...
And a little later :
How did Artin guess his reciprocity
law ? He was not looking for it, not
trying to solve a Hilbert problem.
Neither was he, as would seem so
natural to us today, seeking a
canonical isomorphism, to make
Takagi's theory more functorial. He was led to the law by trying to show...
Read him.
2. Here is a toy example --- not unrelated to class field theory --- of how a bijection can be more natural than others. Let $p$ be a prime number and let $K$ be finite extension of $\mathbb{Q}_p$ containing a primitive $p$-th root of $1$. There are only finitely many degree-$p$ cyclic extensions $L|K$, and there are only finitely many vectorial lines in the $\mathbb{F}_p$-space $K^\times/K^{\times p}$. In fact the two sets have the same number of elements, but the only natural bijection is
$$
L\mapsto\operatorname{Ker}(K^\times/K^{\times p}\to L^\times/L^{\times p}),
$$
of which the reciprocal bijections can be written $D\mapsto K(\root p\of D)$.
It follows that the number of degree-$p$ cyclic extensions $L|K$ is the same as the number of hyperplanes in $K^\times/K^{\times p}$. But is there a natural bijection between these two sets ? You will agree that $L\mapsto N_{L|K}(L^\times)/K^{\times p}$ is as natural a bijection as there can be.
One last point : Given a hyperplane $H\subset K^\times/K^{\times p}$, how do you recover the degree-$p$ cyclic extension $L|K$ such that $H=N_{L|K}(L^\times)/K^{\times p}$ ? Answer : use the natural reciprocity isomorphism $K^\times/K^{\times p}\to\operatorname{Gal}(M|K)$, where $M|K$ is the maximal elementary abelian $p$-extension, to identify $H$ with a subgroup of $\operatorname{Gal}(M|K)$, and take $L=M^H$.
Addendum (2011/11/21) In Recountings (edited by Joel Segel, A K Peters Ltd, Natick, Mass.), Arthur Mattuck recounts a conversation with Emil Artin about his reciprocity law:
I will tell you a story about the
Reciprocity Law. After my thesis, I
had the idea to define $L$-series for
non-abelian extensions. But for them
to agree with the $L$-series for
abelian extensions, a certain
isomorphism had to be true. I could
show it implied all the standard
reciprocity laws. So I called it the
General Reciprocity Law and tried to
prove it but couldn't, even after many
tries. Then I showed it to the other
number theorists, but they all laughed
at it, and I remember Hasse in
particular telling me it couldn't
possibly be true.
Still, I kept at it, but nothing I
tried worked. Not a week went by ---
for three years ! --- that I did not try to prove the Reciprocity Law. It
was discouraging, and meanwhile I
turned to other things. Then one
afternoon I had nothing special to do, so
I said, `Well, I try to prove the
Reciprocity Law again.' So I went out
and sat down in the garden. You see,
from the very beginning I had the idea
to use the cyclotomic fields, but they
never worked, and now I suddenly saw
that all this time I had been using
them in the wrong way --- and in half
an hour I had it.
