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Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" (here). He's working with (φ, Γ)-modules over Q_p not F_p though. But he shows that every one-dimensional (φ, Γ)-module over Q_p is given by a p-adic character δ of $\mathbf{Q}_p^\times$ with the action of φ on a basis vector given by δ(p) and the action of $\gamma\in\Gamma$ given by $\delta(\chi(\gamma))$ (with $\chi$ the $p$-adic cyclotomic character). See section 2 and proposition 3.1 in particular of Colmez's paper (in which he also describes the $H^1$'s of these one-dimensional $(\phi,\Gamma)$-modules).