Let G be the Galois group of Q-bar over Q. I have a good idea of what $H^1(G,GL_1)$ is when $G$ acts trivially on $GL_1$: it is the group of homomorphisms from $G$ to $GL_1$. In particular it factors through the abelian quotient of $G$. In fact it is the group Pontrjagin dual to that abelian quotient.
What if $G$ acts nontrivially on $GL_1$? The automorphisms of $GL_1$ are $x$ and $x^{-1}$. To give a nontrivial action of $G$ on $GL_1$, is the same as giving a map from $G$ to that group with two elements. That's the same as giving a quadratic extension of Q, say $Q(\sqrt{z})$. I'll write $GL_{1,\sqrt{z}}$ for $GL_1$ with that Galois module structure.
How can I compute $H^1(G,GL_{1,\sqrt{z}})$? Is it the Pontrjagin dual of the abelian quotient of the Galois group of Q-bar over $Q(\sqrt{z})$?
Let $\rho : G \to \mathbb{Z}_2$ be a nontrivial homomorphism, and let $N$ denote its kernel (an open normal subgroup of index $2$). Let me write $A$ for the coefficient module; I'll assume that in all of the examples you care about this is abelian. In this setting we can apply the inflation-restriction exact sequence, which here reads
$$0 \to H^1(\mathbb{Z}_2, A) \to H^1(G, A) \to H^1(N, A)^{\mathbb{Z}_2} \to H^2(\mathbb{Z}_2, A) \to H^2(G, A).$$
(We have $A = A^N$ because, by hypothesis, the action of $G$ on $A$ factors through $G/N$.) $H^1(N, A)$ is just homomorphisms $N \to A$, or equivalently $N/[N, N] \to A$, since the action of $N$ on $A$ is trivial, but as you can see there are various other things going on.