So generally one defines the Hopf invariant of a map $f: S^{2n-1} \to S^n$ as the coefficient $H(f)$ in $\alpha^2 = H(f) \beta$ where $\langle \alpha \rangle = H^n(C_f)$ and $\langle \beta \rangle = H^{2n}(C_f)$.
I've looked around and asked a topologist or two, but I can't find any references to the same construction with coefficients. e.g. $\mathbb{Z}_2$ coefficients for instance. The nice bit about this would be that you can extend the statment that the Hopf invariant is 1 for all Hopf fibrations including the trivial one $S^0 \to S^1 \to \mathbb{RP}^1$. Where under normal considerations this would have Hopf invariant 0.
Does anyone know any reference to such a thing or is it possible that this offers no new information? It just seems like a natural extension of the concept, and in general having multiple coefficients you can work with often can differentiate things you couldn't before. I'm only blindly poking in the dark here but I found this interesting.
You cannot really obtain any new information.
Note that for cup product to be defined, you need the coefficient group $k$ to be a ring. Since $\mathbb{Z}$ is initial in rings, there is exactly one map $\mathbb{Z} \rightarrow k$ and this will induce a natural transformation $\phi: H^{*}(-, \mathbb{Z}) \rightarrow H^{*}(-, k)$ of cohomology theories (the "change of coefficients").
Moreover, if you denote by $\alpha, \beta$ the canonical generators of $H^{n}(-, \mathbb{Z}), H^{2n}(-, \mathbb{Z})$, then $\alpha _{k} = \phi(\alpha), \beta _{k} = \phi(\beta)$ will be canonical generators of $H^{n}(-, k), H^{2n}(-, k)$. (To see this one might look at cellular complexes.) Therefore the equation $\alpha ^{2} = H(f) \beta$ will give you
$\alpha _{k} ^{2} = H (f) \beta _{k}$
and so the "Hopf invariant over $k$" will be an integer and in fact equal to $H(f)$.