  Let p be a prime and F a totally real field in which p is unramified. We
consider mod p Hilbert modular forms for F, defined as sections of automorphic
line bundles on Hilbert modular varieties of level prime to p in characteristic
p. For a mod p Hilbert modular Hecke eigenform of arbitrary weight (without
parity hypotheses), we associate a two-dimensional representation of the
absolute Galois group of F, and we give a conjectural description of the set of
weights of all eigenforms from which it arises. This conjecture can be viewed
as a "geometric" variant of the "algebraic" Serre weight conjecture of
Buzzard-Diamond-Jarvis, in the spirit of Edixhoven's variant of Serre's
original conjecture in the case F = Q. We develop techniques for studying the
set of weights giving rise to a fixed Galois representation, and prove results
in support of the conjecture, including cases of partial weight one.
