Hey Rob! You may want to take a look at Colmez's "Représentations triangulines de dimension 2" ([here][1]). He's working with (&phi;, &Gamma;)-modules over **Q**<sub>*p*</sub> not **F**<sub>*p*</sub> though. But he shows that every one-dimensional (&phi;, &Gamma;)-module over **Q**<sub>*p*</sub> is given by a *p*-adic character &delta; of $\mathbf{Q}_p^\times$ with the action of &phi; on a basis vector given by &delta;(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).

EDIT: looking at Colmez's paper, it looks like the way that he knows that this is all one-dimensional $(\phi,\Gamma)$-modules is by using the equivalence of catgeories with Galois representations, so I guess this doesn't really answer your question as it doesn't address the $(\phi,\Gamma)$-modules intrinsically.

  [1]: http://www.math.jussieu.fr/~colmez/Trianguline-ast.pdf