Motivating the coefficient field of $\ell$-adic cohomology It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over $\mathbb{F}_p$ would allow for a possible solution to the Weil conjectures. 
It was also understood that such a cohomology theory could not take values in vector spaces over either the rational numbers $\mathbb{Q}$ or the $p$-adic numbers $\mathbb{Q}_p$. The classical argument that I know is that the cohomology of a supersingular elliptic curve should admit an action by a quaternion algebra structure of the endomorphism group of the elliptic curve, and representation theory rules out such an action. (Side question: Due to whom is this reasoning, and was it known to Weil's generation?)
This reasoning does not, however, rule out the possibility of cohomology with coefficients in $\mathbb{Q}_\ell$ with $\ell \neq p$, and indeed $\ell$-adic cohomology would end up being found. 
Question. Were there any other historical reasons to believe that looking at cohomology valued in $\mathbb{Q}_\ell$-vector spaces would lead to something fruitful?
N.B. This question was posted on MathSE, but I got advised to post it here instead.
 A: One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the $\ell$-adic Tate module of the Jacobian (i.e., the projective limit of $\ell$-power torsion).  Furthermore, if such a curve is defined over a subfield of the complex numbers, then the rank of the Tate module is equal to the rank of the classical degree 1 cohomology of the complexified curve.  We now know that the Tate module is naturally dual to étale cohomology in degree 1.
One might then reasonably hope for a similar relationship between the higher degree classical cohomology of higher dimensional varieties and certain $\ell$-adic Galois representations naturally attached to the variety.
A: I think an important motivation for the $\ell$-adic theory comes from the Riemann existence theorem/the Grauert–Remmert theorem. This says that a finite (topological) covering space $Y \to X$ of a normal complex variety can again be equipped with an algebraic structure, which is more or less what you need to prove to obtain
$$\pi_1^{\operatorname{\acute et}}(X) = \widehat{\pi_1^{\operatorname{top}}(X(\mathbf C))}.$$
So finite covering spaces can be detected using the étale topology (except it's not really a topology, but that's not stopping Grothendieck!).
In particular, you expect to get a good theory of étale cohomology with finite coefficients. But to run the arguments that Weil was dreaming of, you need characteristic $0$ coefficients, so what do you do? Just take a limit!
I think that's really the explanation of why the adic formalism enters the picture. But it doesn't quite explain why it's different at the prime $p$. There are a few ways to look at this:

*

*Serre's argument shows that a $\mathbf Q_p$-valued Weil cohomology theory cannot exist (over any ground field $k$ containing $\mathbf F_{p^2}$; so in particular for $k = \bar{\mathbf F}_p$).

*The basic results on $\ell$-adic étale cohomology take the Galois cohomology of function fields of curves over algebraically closed fields as a starting point, and these behave differently at the prime $p$.

*Already for elliptic curves, the $p$-adic Tate module behaves a little different from the $\ell$-adic one.

In the end it doesn't really matter, because all they needed was one Weil cohomology theory.
