Tate writes in his article in Cassels and Frolich that the cohomology calculations have their origin in genus theory (going back --- at least --- to Gauss's *Disquitiones*).

I haven't gone that far back, but I have read parts of Hilbert's *Zahlbericht*, where e.g. his Theorem 90 appeared.
(The theorems are labelled in order throughout the book.)

If you look at how Hilbert argues there, he uses Theorem 90 in much the same way
we do --- it gives a vanishing result, which can be fed into other computations via
the snake lemma (i.e. the boundary map in the cohomology long exact sequence). Indeed, he has many calculations that are concrete forms of the snake lemma in the
book.

There are also computations with units, especially in the case of cyclic extensions,
which are analogous to what we would describe as computing the Herbrand quotient
of the unit group (which nowadays appears as an input along the way to computing
the Herbrand quotient of the idele class group for a cyclic extension). (Note
that these sorts of computations go back to Kummer. The chapter by Rosen in
*Modular Forms and Fermat's Last Theorem* gives some nice insight into this.)

As Rene Schipperus notes in another answer, there was a (at least apparently)
separate thread in the first half of the 20th century, in which Brauer, Hasse,
and Noether were studying central simple algebras over number fields,
and discovered and proved that the sum of the local invariants is zero. They
then realized that this result could be applied to reprove Artin's reciprocity
law. (See this historical survey for more details.)

When the theory of central simple algebras was recast cohomologically, this
development was unified with the earlier arguments of genus theory, and
the modern cohomological treatment emerged.

(There is yet another thread of development, which is the replacement
of $L$-functions in the argument with algebraic arguments via Kummer theory,
which isn't quite as relevant to your question.)

Genus theory isn't so commonly treated in modern alg. no. theory books.
*Advanced number theory*, by Harvey Cohn, gives a discussion for quadratic
fields (although the cohomological aspects are not made explicit, you can
see that Hilbert's Thm. 90 plays an important role). Here is an exercise sheet that turns up when
you google genus theory, and which is more cohomological.

[Incidentally, while group cohomology literally came out of alg. topology,
the computations in alg. number theory that it captures are much older!
E.g. Hilbert's Thm. 90, at least in the cyclic case, certainly predates the invention of group cohomology by algebraic topologists. One thing that
group cohomology is good at is expressing the non-cylic case; my memory
is that there was a period when people --- maybe Emmy Noether --- expressed
the non-cyclic case of HT90 in the language of crossed homomorphisms etc.
that can be used to talk about group $H^1$. It must have been a relief
to discover that all the different computations, crossed homomorphisms,
factor sets, local invariants, proofs of the Artin rec. law, and so on,
could be unified by the language of gp. cohomology!]